OFFSET
0,4
COMMENTS
Second diagonal is A000567.
FORMULA
E.g.f.: exp(t*x/(1-3*x))/(1-3*x)^(1/3).
EXAMPLE
Taylor series starts:
1 + (t + 1)*x + (t^2 + 8*t + 4)*x^2 + (t^3 + 21*t^2 + 84*t + 28)*x^3 + (t^4 + 40*t^3 + 420*t^2 + 1120*t + 280)*x^4 + O(x^5).
Triangle starts:
[0] 1
[1] 1, 1
[2] 4, 8, 1
[3] 28, 84, 21, 1
[4] 280, 1120, 420, 40, 1
[5] 3640, 18200, 9100, 1300, 65, 1
[6] 58240, 349440, 218400, 41600, 3120, 96, 1
[7] 1106560, 7745920, 5809440, 1383200, 138320, 6384, 133, 1
[8] 24344320, 194754560, 170410240, 48688640, 6086080, 374528, 11704, 176, 1
MAPLE
ser := n -> series(KummerU(-n, 1/3, x), x, n+1):
seq(seq((-3)^(n-k)*coeff(ser(n), x, k), k=0..n), n=0..8);
# Alternative:
gf := exp(t*x/(1-3*x))/(1-3*x)^(1/3): ser := n -> series(gf, x, n+1):
c := n -> coeff(ser(n), x, n): seq(seq(n!*coeff(c(n), t, k), k=0..n), n=0..8);
MATHEMATICA
(* rows[n], n[0..oo] *)
n=12; r={}; For[k=0, k<n+1, k++, AppendTo[r, Binomial[n, n-k]/Product[3*j+1, {j, 0, k-1}]*Product[3*j+1, {j, 0, n-1}]]]; r
(* columns[k], k[0..oo] *)
k=2; c={}; For[n=k, n<13, n++, AppendTo[c, Binomial[n, n-k]/Product[3*j+1, {j, 0, k-1}]*Product[3*j+1, {j, 0, n-1}]]]; c
(* sequence *)
s={}; For[n=0, n<13, n++, For[k=0, k<n+1, k++, AppendTo[s, Binomial[n, n-k]/Product[3*j+1, {j, 0, k-1}]*Product[3*j+1, {j, 0, n-1}]]]]; s
(* Detlef Meya, Jul 31 2023 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Jan 18 2020
STATUS
approved