A331331 - OEIS

%I #14 Sep 03 2023 10:14:10

%S 1,1,1,4,8,1,28,84,21,1,280,1120,420,40,1,3640,18200,9100,1300,65,1,

%T 58240,349440,218400,41600,3120,96,1,1106560,7745920,5809440,1383200,

%U 138320,6384,133,1,24344320,194754560,170410240,48688640,6086080,374528,11704,176,1

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

%C Second diagonal is A000567.

%F E.g.f.: exp(t*x/(1-3*x))/(1-3*x)^(1/3).

%e Taylor series starts:

%e 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).

%e Triangle starts:

%e [0] 1

%e [1] 1,        1

%e [2] 4,        8,         1

%e [3] 28,       84,        21,        1

%e [4] 280,      1120,      420,       40,       1

%e [5] 3640,     18200,     9100,      1300,     65,      1

%e [6] 58240,    349440,    218400,    41600,    3120,    96,     1

%e [7] 1106560,  7745920,   5809440,   1383200,  138320,  6384,   133,   1

%e [8] 24344320, 194754560, 170410240, 48688640, 6086080, 374528, 11704, 176, 1

%p ser := n -> series(KummerU(-n, 1/3, x), x, n+1):

%p seq(seq((-3)^(n-k)*coeff(ser(n), x, k), k=0..n), n=0..8);

%p # Alternative:

%p gf := exp(t*x/(1-3*x))/(1-3*x)^(1/3): ser := n -> series(gf, x, n+1):

%p c := n -> coeff(ser(n), x, n): seq(seq(n!*coeff(c(n), t, k), k=0..n), n=0..8);

%t (* rows[n], n[0..oo] *)

%t 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

%t (* columns[k], k[0..oo] *)

%t 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

%t (* sequence *)

%t 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

%t (* _Detlef Meya_, Jul 31 2023 *)

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

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

%K nonn,tabl

%O 0,4

%A _Peter Luschny_, Jan 18 2020