A176157 Regular triangle, T(n, k) = f(n, k) - f(n, 0) + 1, where 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), read by rows.

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

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

Cf. A008277, A176153, A176154, A176155, A176156.

Sequence in context: A267751 A185996 A261215 * A176156 A172339 A342972

Adjacent sequences: A176154 A176155 A176156 * A176158 A176159 A176160

Keyword

sign,tabl

Author

Roger L. Bagula, Apr 10 2010

Extensions

Name edited by G. C. Greubel, Nov 26 2019

Status

approved