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

%I A331331 #14 Sep 03 2023 10:14:10
%S A331331 1,1,1,4,8,1,28,84,21,1,280,1120,420,40,1,3640,18200,9100,1300,65,1,
%T A331331 58240,349440,218400,41600,3120,96,1,1106560,7745920,5809440,1383200,
%U A331331 138320,6384,133,1,24344320,194754560,170410240,48688640,6086080,374528,11704,176,1
%N A331331 Triangle read by rows, T(n, k) (0 <= k <= n) = (-m)^(n-k)*[x^k] KummerU(-n, 1/m, x) for m = 3.
%C A331331 Second diagonal is A000567.
%F A331331 E.g.f.: exp(t*x/(1-3*x))/(1-3*x)^(1/3).
%e A331331 Taylor series starts:
%e A331331 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 A331331 Triangle starts:
%e A331331 [0] 1
%e A331331 [1] 1,        1
%e A331331 [2] 4,        8,         1
%e A331331 [3] 28,       84,        21,        1
%e A331331 [4] 280,      1120,      420,       40,       1
%e A331331 [5] 3640,     18200,     9100,      1300,     65,      1
%e A331331 [6] 58240,    349440,    218400,    41600,    3120,    96,     1
%e A331331 [7] 1106560,  7745920,   5809440,   1383200,  138320,  6384,   133,   1
%e A331331 [8] 24344320, 194754560, 170410240, 48688640, 6086080, 374528, 11704, 176, 1
%p A331331 ser := n -> series(KummerU(-n, 1/3, x), x, n+1):
%p A331331 seq(seq((-3)^(n-k)*coeff(ser(n), x, k), k=0..n), n=0..8);
%p A331331 # Alternative:
%p A331331 gf := exp(t*x/(1-3*x))/(1-3*x)^(1/3): ser := n -> series(gf, x, n+1):
%p A331331 c := n -> coeff(ser(n), x, n): seq(seq(n!*coeff(c(n), t, k), k=0..n), n=0..8);
%t A331331 (* rows[n], n[0..oo] *)
%t A331331 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 A331331 (* columns[k], k[0..oo] *)
%t A331331 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 A331331 (* sequence *)
%t A331331 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 A331331 (* _Detlef Meya_, Jul 31 2023 *)
%Y A331331 Cf. T(n, 0) = A007559(n), T(n, n-1) = A000567(n) for n >= 1.
%Y A331331 Cf. |A021009| (m=1), A176230 (m=2), this sequence (m=3).
%K A331331 nonn,tabl
%O A331331 0,4
%A A331331 _Peter Luschny_, Jan 18 2020