A331331 Triangle read by rows, T(n, k) (0 <= k <= n) = (-m)^(n-k)*[x^k] KummerU(-n, 1/m, x) for m = 3.

1, 1, 1, 4, 8, 1, 28, 84, 21, 1, 280, 1120, 420, 40, 1, 3640, 18200, 9100, 1300, 65, 1, 58240, 349440, 218400, 41600, 3120, 96, 1, 1106560, 7745920, 5809440, 1383200, 138320, 6384, 133, 1, 24344320, 194754560, 170410240, 48688640, 6086080, 374528, 11704, 176, 1

Offset

0,4

Comments

Second diagonal is A000567.

Links

Table of n, a(n) for n=0..44.

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

Cf. T(n, 0) = A007559(n), T(n, n-1) = A000567(n) for n >= 1.

Cf. |A021009| (m=1), A176230 (m=2), this sequence (m=3).

Sequence in context: A340423 A367416 A295086 * A134484 A244641 A274192

Adjacent sequences: A331328 A331329 A331330 * A331332 A331333 A331334

Keyword

nonn,tabl

Author

Peter Luschny, Jan 18 2020

Status

approved