A176153 -id:A176153 - cp's OEIS frontend

A176154 Triangle T(n,k) = Sum_{j=0..k} Stirling1(n, n-j)binomial(n,j) + Sum_{j=0..n-k} Stirling1(n, n-j)binomial(n,j), read by rows.

2, 2, 2, 0, -2, 0, -1, -10, -10, -1, 20, -4, 86, -4, 20, -78, -128, 102, 102, -128, -78, 77, -13, 1982, -6628, 1982, -13, 77, 2641, 2494, 1129, 12448, 12448, 1129, 2494, 2641, -36944, -37168, 12168, -463496, 745726, -463496, 12168, -37168, -36944

Offset: 0

Roger L. Bagula, Apr 10 2010

			Triangle begins as:
     2;
     2,    2;
     0,   -2,    0;
    -1,  -10,  -10,    -1;
    20,   -4,   86,    -4,    20;
   -78, -128,  102,   102,  -128,  -78;
    77,  -13, 1982, -6628,  1982,  -13,   77;
  2641, 2494, 1129, 12448, 12448, 1129, 2494, 2641;

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A048994, A132393, A176153, A176155, A176156, A176157.

T:= function(n,k) return Sum([0..k], j-> (-1)^j*Stirling1(n,n-j)*Binomial(n,j)) + Sum([0..n-k], j-> (-1)^j*Stirling1(n, n-j)*Binomial(n,j)); end;
Flat(List([0..10], n-> List([0..n], k-> T(n, k) ))); # G. C. Greubel, Nov 26 2019

T:= func< n,k | (&+[StirlingFirst(n,n-j)*Binomial(n,j): j in [0..k]]) + (&+[StirlingFirst(n,n-j)*Binomial(n,j): j in [0..n-k]]) >;
[T(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 26 2019

with(combinat);
T:= proc(n, k) option remember; add(stirling1(n, n-j)*binomial(n, j), j=0..k) + add(stirling1(n, n-j)* binomial(n, j), j=0..n-k); end;
seq(seq(T(n,k), k=0..n), n=0..10); # G. C. Greubel, Nov 26 2019

T[n_, k_]:= Sum[StirlingS1[n, n-j]*Binomial[n, j], {j, 0, k}] + Sum[StirlingS1[n, n-j]*Binomial[n, j], {j, 0, n-k}];
Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten

T(n,k) = sum(j=0,k, stirling(n,n-j,1)*binomial(n,j)) + sum(j=0,n-k, stirling(n, n-j,1)*binomial(n,j)); \\ G. C. Greubel, Nov 26 2019

def T(n, k): return sum((-1)^j*stirling_number1(n,n-j)*binomial(n,j) for j in (0..k)) + sum((-1)^j*stirling_number1(n, n-j)*binomial(n,j) for j in (0..n-k))
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Nov 26 2019

T(n, k) = Sum_{j=0..k} Stirling1(n, n-j)*binomial(n,j) + Sum_{j=0..n-k} Stirling1(n, n-j)*binomial(n,j).

Name edited by G. C. Greubel, Nov 27 2019

A176155 Regular triangle, T(n, k) = f(n, k) - f(n, 0) + 1, where f(n, k) = Sum_{j=0..k} StirlingS1(n, n-j)binomial(n,j) + Sum_{j=0..n-k} StirlingS1(n, n-j)binomial(n, j), read by rows.

Original entry on oeis.org

1, 1, 1, 1, -1, 1, 1, -8, -8, 1, 1, -23, 67, -23, 1, 1, -49, 181, 181, -49, 1, 1, -89, 1906, -6704, 1906, -89, 1, 1, -146, -1511, 9808, 9808, -1511, -146, 1, 1, -223, 49113, -426551, 782671, -426551, 49113, -223, 1, 1, -323, -343547, 3220453, -3873389, -3873389, 3220453, -343547, -323, 1

Offset: 0

Roger L. Bagula, Apr 10 2010

Row sum are: {1, 2, 1, -14, 23, 266, -3068, 16304, 27351, -1993610, 31213301, ...}.

			Triangle begins as:
  1;
  1,    1;
  1,   -1,       1;
  1,   -8,      -8,       1;
  1,  -23,      67,     -23,        1;
  1,  -49,     181,     181,      -49,        1;
  1,  -89,    1906,   -6704,     1906,      -89,       1;
  1, -146,   -1511,    9808,     9808,    -1511,    -146,       1;
  1, -223,   49113, -426551,   782671,  -426551,   49113,    -223,    1;
  1, -323, -343547, 3220453, -3873389, -3873389, 3220453, -343547, -323, 1;

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A048994, A132393, A176153, A176154, A176156, A176157.

f:= function(n,k) return Sum([0..k], j-> (-1)^j*Stirling1(n,n-j)*Binomial(n,j)) + Sum([0..n-k], j-> (-1)^j*Stirling1(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

f:= func< n,k | (&+[StirlingFirst(n,n-j)*Binomial(n,j): j in [0..k]]) + (&+[StirlingFirst(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

with(combinat);
f:= proc(n, k) option remember; add(stirling1(n, n-j)*binomial(n, j), j=0..k) + add(stirling1(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

f[n_, k_]:= Sum[StirlingS1[n, n-j]*Binomial[n, j], {j,0,k}] + Sum[StirlingS1[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

f(n,k) = sum(j=0,k, stirling(n,n-j,1)*binomial(n,j)) + sum(j=0,n-k, stirling(n, n-j,1)*binomial(n,j));
T(n,k) = f(n,k) - f(n,0) + 1; \\ G. C. Greubel, Nov 26 2019

def f(n, k): return sum((-1)^j*stirling_number1(n,n-j)*binomial(n,j) for j in (0..k)) + sum((-1)^j*stirling_number1(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

With f(n, k) = Sum_{j=0..k} StirlingS1(n, n-j)*binomial(n,j) + Sum_{j=0..n-k} StirlingS1(n, n-j)*binomial(n, j) then T(n, k) = f(n, k) - f(n, 0) + 1.

Name edited by G. C. Greubel, Nov 27 2019

A176156 Regular triangle, T(n, k) = f(n, k) - f(n, 0) + 1, where f(n, k) = Sum_{j=0..k} (-1)^jStirlingS2(n, n-j)binomial(n,j) + Sum_{j=0..n-k} (-1)^jStirlingS2(n, n-j)binomial(n, j), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 10, 10, 1, 1, 25, 67, 25, 1, 1, 51, 281, 281, 51, 1, 1, 91, 646, 1036, 646, 91, 1, 1, 148, -1217, -12536, -12536, -1217, 148, 1, 1, 225, -31079, -287223, -548785, -287223, -31079, 225, 1, 1, 325, -342899, -3906899, -11000741, -11000741, -3906899, -342899, 325, 1

Offset: 0

Roger L. Bagula, Apr 10 2010

Row sum are: {1, 2, 5, 22, 119, 666, 2512, -27208, -1184937, -30500426, -716845999, ...}.

			Triangle begins as:
  1;
  1,   1;
  1,   3,      1;
  1,  10,     10,       1;
  1,  25,     67,      25,       1;
  1,  51,    281,     281,      51,       1;
  1,  91,    646,    1036,     646,      91,      1;
  1, 148,  -1217,  -12536,  -12536,   -1217,    148,   1;
  1, 225, -31079, -287223, -548785, -287223, -31079, 225, 1;

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A048994, A132393, A176153, A176154, A176155, A176157.

f:= function(n,k) return Sum([0..k], j-> Stirling1(n,n-j)*Binomial(n,j)) + Sum([0..n-k], j-> Stirling1(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

f:= func< n,k | (&+[(-1)^j*StirlingFirst(n,n-j)*Binomial(n,j): j in [0..k]]) + (&+[(-1)^j*StirlingFirst(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

with(combinat);
f:= proc(n, k) option remember; add((-1)^j*stirling1(n, n-j)*binomial(n, j), j=0..k) + add((-1)^j*stirling1(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

f[n_, k_]:= Sum[(-1)^j*StirlingS1[n, n-j]*Binomial[n, j], {j,0,k}] + Sum[(-1)^j*StirlingS1[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

f(n,k) = sum(j=0,k, (-1)^j*stirling(n,n-j,1)*binomial(n,j)) + sum(j=0,n-k, (-1)^j*stirling(n, n-j,1)*binomial(n,j));
T(n,k) = f(n,k) - f(n,0) + 1; \\ G. C. Greubel, Nov 26 2019

def f(n, k): return sum(stirling_number1(n,n-j)*binomial(n,j) for j in (0..k)) + sum(stirling_number1(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

With f(n, k) = Sum_{j=0..k} (-1)^j*StirlingS1(n, n-j)*binomial(n,j) + Sum_{j=0..n-k} (-1)^j*StirlingS1(n, n-j)*binomial(n, j) then T(n, k) = f(n, k) - f(n, 0) + 1.

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

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.

Original entry on oeis.org

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

Roger L. Bagula, Apr 10 2010

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

			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;

G. C. Greubel, Rows n = 0..100 of triangle, flattened

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

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

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

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

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

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

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

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.

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

cp's OEIS Frontend

A176154 Triangle T(n,k) = Sum_{j=0..k} Stirling1(n, n-j)*binomial(n,j) + Sum_{j=0..n-k} Stirling1(n, n-j)*binomial(n,j), read by rows.

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A176155 Regular triangle, T(n, k) = f(n, k) - f(n, 0) + 1, where f(n, k) = Sum_{j=0..k} StirlingS1(n, n-j)*binomial(n,j) + Sum_{j=0..n-k} StirlingS1(n, n-j)*binomial(n, j), read by rows.

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A176156 Regular triangle, T(n, k) = f(n, k) - f(n, 0) + 1, where f(n, k) = Sum_{j=0..k} (-1)^j*StirlingS2(n, n-j)*binomial(n,j) + Sum_{j=0..n-k} (-1)^j*StirlingS2(n, n-j)*binomial(n, j), read by rows.

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Programs

GAP

Magma

Maple

Mathematica

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Extensions

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.

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Author

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Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A176154 Triangle T(n,k) = Sum_{j=0..k} Stirling1(n, n-j)binomial(n,j) + Sum_{j=0..n-k} Stirling1(n, n-j)binomial(n,j), read by rows.

A176155 Regular triangle, T(n, k) = f(n, k) - f(n, 0) + 1, where f(n, k) = Sum_{j=0..k} StirlingS1(n, n-j)binomial(n,j) + Sum_{j=0..n-k} StirlingS1(n, n-j)binomial(n, j), read by rows.

A176156 Regular triangle, T(n, k) = f(n, k) - f(n, 0) + 1, where f(n, k) = Sum_{j=0..k} (-1)^jStirlingS2(n, n-j)binomial(n,j) + Sum_{j=0..n-k} (-1)^jStirlingS2(n, n-j)binomial(n, j), read by rows.

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.