Sequence Daisy

0	1	2	3	4	5	6	7
1	2	5	16	67	374	2825	29212

row sums of Oak

🌊 triangle Oak (A022166) row sums Daisy (A006116)
k n	0	1	2	3	4	5	6	7	sums
0	1								1
1	1	1							2
2	1	3	1						5
3	1	7	7	1					16
4	1	15	35	15	1				67
5	1	31	155	155	31	1			374
6	1	63	651	1395	651	63	1		2825
7	1	127	2667	11811	11811	2667	127	1	29212

💧 triangle OakDrop (~A289537) row sums Dahlia (~A182176)
k n	0	1	2	3	4	5	6	7	sums
0	1								1
1	0	1							1
2	0	2	1						3
3	0	4	6	1					11
4	0	8	28	14	1				51
5	0	16	120	140	30	1			307
6	0	32	496	1240	620	62	1		2451
7	0	64	2016	10416	11160	2604	126	1	26387

row sums of Ash

Pascal with columns multiplied by Aster

PascalDrop with columns multiplied by Aster

💧 triangle Aspen row sums Dahlia (~A182176)
k n	0	1	2	3	4	5	6	7	sums
0	1 · 1 1								1
1	0	1 · 1 1							1
2	0	1 · 1 1	1 · 2 2						3
3	0	1 · 1 1	2 · 2 4	1 · 6 6					11
4	0	1 · 1 1	3 · 2 6	3 · 6 18	1 · 26 26				51
5	0	1 · 1 1	4 · 2 8	6 · 6 36	4 · 26 104	1 · 158 158			307
6	0	1 · 1 1	5 · 2 10	10 · 6 60	10 · 26 260	5 · 158 790	1 · 1330 1330		2451
7	0	1 · 1 1	6 · 2 12	15 · 6 90	20 · 26 520	15 · 158 2370	6 · 1330 7980	1 · 15414 15414	26387

data

	0	1	2	3	4	5	6	7
🌊 Daisy	1	2	5	16	67	374	2825	29212
💧 Dahlia	1	1	3	11	51	307	2451	26387

$\mathrm {Daisy} (n)~~=~~\sum _{i=0}^{n}\mathrm {Dahlia} (i)$
${\begin{matrix}\mathrm {Daisy} (n)&~~=~~&\sum _{i=0}^{n}\mathrm {Dahlia} (i)\\\\\mathrm {Daisy} (5)&~~=~~&\mathrm {Dahlia} (0)~+~...~+~\mathrm {Dahlia} (5)\\374&~~=~~&1+1+3+11+51+307\end{matrix}}$

$\mathrm {Dahlia} (n)~~=~~\mathrm {Daisy} (n-1)~+~(2^{n-1}-1)\cdot \mathrm {Daisy} (n-2)$
${\begin{matrix}\mathrm {Dahlia} (n)&~~=~~&\mathrm {Daisy} (n-1)&~+~&(2^{n-1}-1)&\cdot &\mathrm {Daisy} (n-2)\\\\\mathrm {Dahlia} (6)&~~=~~&\mathrm {Daisy} (5)&~+~&2^{5}-1&\cdot &\mathrm {Daisy} (4)\\2451&~~=~~&374&~+~&31&\cdot &67\end{matrix}}$

ShortDahliaDrop is the product of padded Daisy and the powers of two. Padded means, that there is a 1 before the other entries.