OFFSET
1,1
COMMENTS
REFERENCES
B. Harris and C J. Park, A generalization of Eulerian numbers with a probabilistic Application, Statistics and Probability Letters 20 (1994), page 40
LINKS
G. C. Greubel, Rows n=1..100 of triangle, flattened
FORMULA
T(n,k) = Sum_{j=0..k} (-1)^j *binomial(n+1,j)*(k+2-j)^n, for k=0..n-1.
EXAMPLE
Triangle begins with:
2;
4, -3;
8, -5, 4;
16, 1, 11, -5;
32, 51, 46, -14, 6;
64, 281, 337, 22, 22, -7;
128, 1163, 2472, 1121, 176, -27, 8;
256, 4257, 15703, 15493, 4419, 163, 37, -9;
512, 14563, 88354, 155980, 88486, 14398, 622, -44, 10;
1024, 47785, 455357, 1310024, 1310816, 454730, 48170, 848, 56, -11;
...
MAPLE
A180246 := proc(n, k) add( (-1)^v*binomial(n+1, v)*(k+2-v)^n, v=0..k) ; end proc: # R. J. Mathar, Jan 29 2011
P := proc(n, x) option remember; if n = 0 then 1 else
(n*x+2*(1-x))*P(n-1, x)+x*(1-x)*diff(P(n-1, x), x);
expand(%) fi end:
A180246 := (n, k) -> coeff(P(n, x), x, k):
seq(print(seq(A180246(n, k), k=0..n-1)), n=0..10); # Peter Luschny, Mar 07 2014
MATHEMATICA
t[n_, j_, d_]:= Sum[(-1)^v *Binomial[n+1, v](j+d-v)^n, {v, 0, j}];
Table[Flatten[Table[Table[t[n, k, m], {k, 0, n-1}], {n, 1, 10}]], {m, 0, 10}]
(* This sequence corresponds to m=2 *)
Table[Sum[(-1)^j*Binomial[n+1, j]*(k-j+2)^n, {j, 0, k}], {n, 1, 12}, {k, 0, n-1}]//Flatten
PROG
(PARI) {T(n, k) = sum(j=0, k, (-1)^j*binomial(n+1, j)*(k-j+2)^n)};
for(n=1, 12, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Feb 23 2019
(Magma) [[(&+[(-1)^j*Binomial(n+1, j)*(k-j+2)^n: j in [0..k]]): k in [0..n-1]]: n in [1..12]]; // G. C. Greubel, Feb 23 2019
(Sage) [[sum((-1)^j*binomial(n+1, j)*(k-j+2)^n for j in (0..k)) for k in (0..n-1)] for n in (1..12)] # G. C. Greubel, Feb 23 2019
(GAP) Flat(List([1..12], n-> List([0..n-1], k-> Sum([0..k], j-> (-1)^j*Binomial(n+1, j)*(k-j+2)^n )))); # G. C. Greubel, Feb 23 2019
CROSSREFS
KEYWORD
AUTHOR
Roger L. Bagula, Aug 19 2010
STATUS
approved