OFFSET
0,5
COMMENTS
Row sum are: {1, 2, 5, 22, 115, 696, 4699, 34476, 269483, 2198128, 18229726, ...}.
The first negative terms are T(14,6) = T(14,8) = -17062199622 = a(111), T(14,7) = -38263538781, T(15,5) = T(15,10) = -18803914339, T(15,6) = T(15,9) = -315758882649, T(15,7) = T(15,8) = -1027328563614. - Georg Fischer, Hugo Pfoertner, Jul 16 2020
LINKS
G. C. Greubel, Rows n = 0..100 of triangle, flattened
FORMULA
With f(n, k) = Sum_{j=0..k} StirlingS2(n, n-j)*binomial(n,j) + Sum_{j=0..n-k} StirlingS2(n, n-j)*binomial(n, j) then T(n, k) = f(n, k) - f(n, 0) + 1.
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 3, 1;
1, 10, 10, 1;
1, 25, 63, 25, 1;
1, 51, 296, 296, 51, 1;
1, 91, 1060, 2395, 1060, 91, 1;
1, 148, 3081, 14008, 14008, 3081, 148, 1;
1, 225, 7665, 62909, 127883, 62909, 7665, 225, 1;
1, 325, 16948, 230032, 851758, 851758, 230032, 16948, 325, 1;
1, 451, 34191, 716796, 4390866, 7945116, 4390866, 716796, 34191, 451, 1;
MAPLE
with(combinat);
f:= proc(n, k) option remember; add(stirling2(n, n-j)*binomial(n, j), j=0..k) + add(stirling2(n, n-j)* binomial(n, j), j=0..n-k); end;
seq(seq(f(n, k) -f(n, 0) +1, k=0..n), n=0..10); # G. C. Greubel, Nov 26 2019
MATHEMATICA
f[n_, k_]:= Sum[StirlingS2[n, n-j]*Binomial[n, j], {j, 0, k}] + Sum[StirlingS2[n, n-j]*Binomial[n, j], {j, 0, n-k}];
Table[f[n, k] - f[n, 0] + 1, {n, 0, 10}, {k, 0, n}]//Flatten
PROG
(PARI)
f(n, k) = sum(j=0, k, stirling(n, n-j, 2)*binomial(n, j)) + sum(j=0, n-k, stirling(n, n-j, 2)*binomial(n, j));
T(n, k) = f(n, k) - f(n, 0) + 1; \\ G. C. Greubel, Nov 26 2019
(Magma)
f:= func< n, k | (&+[StirlingSecond(n, n-j)*Binomial(n, j): j in [0..k]]) + (&+[StirlingSecond(n, n-j)*Binomial(n, j): j in [0..n-k]]) >;
[f(n, k) - f(n, 0) + 1: k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 26 2019
(Sage)
def f(n, k): return sum(stirling_number2(n, n-j)*binomial(n, j) for j in (0..k)) + sum(stirling_number2(n, n-j)*binomial(n, j) for j in (0..n-k))
[[f(n, k)-f(n, 0)+1 for k in (0..n)] for n in (0..10)] # G. C. Greubel, Nov 26 2019
(GAP)
f:= function(n, k) return Sum([0..k], j-> Stirling2(n, n-j)*Binomial(n, j)) + Sum([0..n-k], j-> Stirling2(n, n-j)*Binomial(n, j)); end;
Flat(List([0..10], n-> List([0..n], k-> f(n, k)-f(n, 0)+1 ))); # G. C. Greubel, Nov 26 2019
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Roger L. Bagula, Apr 10 2010
EXTENSIONS
Name edited by G. C. Greubel, Nov 26 2019
STATUS
approved