A154252 -id:A154252 - cp's OEIS frontend

A168620 Table T(n,k) with the coefficient [x^k] of the polynomial 5(x+1)^n - 4(x^n+1) in column 0<=k<=n. T(0,0)=1.

1, 1, 1, 1, 10, 1, 1, 15, 15, 1, 1, 20, 30, 20, 1, 1, 25, 50, 50, 25, 1, 1, 30, 75, 100, 75, 30, 1, 1, 35, 105, 175, 175, 105, 35, 1, 1, 40, 140, 280, 350, 280, 140, 40, 1, 1, 45, 180, 420, 630, 630, 420, 180, 45, 1, 1, 50, 225, 600, 1050, 1260, 1050, 600, 225, 50, 1

Offset: 0

Roger L. Bagula and Gary W. Adamson, Dec 01 2009

Row sums are apparently in A154252.

			1;
1, 1;
1, 10, 1;
1, 15, 15, 1;
1, 20, 30, 20, 1;
1, 25, 50, 50, 25, 1;
1, 30, 75, 100, 75, 30, 1;
1, 35, 105, 175, 175, 105, 35, 1;
1, 40, 140, 280, 350, 280, 140, 40, 1;
1, 45, 180, 420, 630, 630, 420, 180, 45, 1;
1, 50, 225, 600, 1050, 1260, 1050, 600, 225, 50, 1;

m = 2; p[x_, n_] := If[n == 0, 1, (2*m + 1)(x + 1)^n - 2*m*(x^n + 1)]
a = Table[CoefficientList[p[x, n], x], {n, 0, 10}];
Flatten[a]

A154312 Triangle T(n,k), 0<=k<=n, read by rows, given by [0,1/2,-1/2,0,0,0,0,0,0,0,...] DELTA [2,-1/2,-1/2,2,0,0,0,0,0,0,0 ...] where DELTA is the operator defined in A084938 .

Original entry on oeis.org

1, 0, 2, 0, 1, 3, 0, 0, 3, 5, 0, 0, 0, 7, 9, 0, 0, 0, 0, 15, 17, 0, 0, 0, 0, 0, 31, 33, 0, 0, 0, 0, 0, 0, 63, 65, 0, 0, 0, 0, 0, 0, 0, 127, 129, 0, 0, 0, 0, 0, 0, 0, 0, 255, 257, 0, 0, 0, 0, 0, 0, 0, 0, 0, 511, 513, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1023, 1025, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2047

Offset: 0

Philippe Deléham, Jan 07 2009

Column sums give A003945.

			Triangle begins:
1;
0, 2;
0, 1, 3;
0, 0, 3, 5;
0, 0, 0, 7, 9;
0, 0, 0, 0, 15, 17; ...

Cf. A000225, A094373

Sum_{k, 0<=k<=n}T(n,k)*x^(n-k)= A040000(n), A094373(n), A000079(n), A083329(n), A095121(n), A154117(n), A131128(n), A154118(n), A131130(n), A154251(n), A154252(n) for x = -1,0,1,2,3,4,5,6,7,8,9 respectively.
G.f.: (1-x*y+x^2*y-x^2*y^2)/(1-3*x*y+2*x^2*y^2). - Philippe Deléham, Nov 02 2013
T(n,k) = 3*T(n-1,k-1) - 2*T(n-2,k-2), T(0,0) = 1, T(1,0) = 0, T(1,1) = 2, T(2,0) = 0, T(2,1) = 1, T(2,2) = 3, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Nov 02 2013

A168621 Triangle read by rows: T(n,0) = T(n,n) = 1 for n >= 0, T(n,k) = ((n - 1)! + 1)*binomial(n, k) for 1 <= k <= n - 1, n >= 2.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 9, 9, 1, 1, 28, 42, 28, 1, 1, 125, 250, 250, 125, 1, 1, 726, 1815, 2420, 1815, 726, 1, 1, 5047, 15141, 25235, 25235, 15141, 5047, 1, 1, 40328, 141148, 282296, 352870, 282296, 141148, 40328, 1, 1, 362889, 1451556, 3386964, 5080446, 5080446, 3386964, 1451556, 362889, 1

Offset: 0

Roger L. Bagula and Gary W. Adamson, Dec 01 2009

Row 0 is 1, and row n gives the coefficients in the expansion of (x + 1)^n + (n - 1)!*((x + 1)^n - x^n -1). - Franck Maminirina Ramaharo, Dec 22 2018

			Triangle begins:
  1;
  1,     1;
  1,     4,      1;
  1,     9,      9,      1;
  1,    28,     42,     28,      1;
  1,   125,    250,    250,    125,      1;
  1,   726,   1815,   2420,   1815,    726,      1;
  1,  5047,  15141,  25235,  25235,  15141,   5047,     1;
  1, 40328, 141148, 282296, 352870, 282296, 141148, 40328, 1;
  ...

Row sums: A154252.

Cf. A007318, A132047, A155863, A155864, A155865 A168619, A219570.

p[x_, n_] := If[n == 0, 1, (x + 1)^n + (n - 1)!*((x + 1)^n - x^n - 1)];
Table[CoefficientList[p[x, n], x], {n, 0, 12}] // Flatten (* Franck Maminirina Ramaharo, Dec 22 2018 *)

T(n, k) := if k = 0 or k = n then 1 else ((n - 1)! + 1)*binomial(n, k)$
create_list(T(n, k), n, 0, 12, k, 0, n); /* Franck Maminirina Ramaharo, Dec 22 2018 */

From Franck Maminirina Ramaharo, Dec 22 2018: (Start)
T(n,k) = A007318(n,k) + A219570(n,k) for 1 <= k <= n - 1, n >= 2.
E.g.f.: exp((1 + x)*y) + log((1 - y)*(1 - x*y)/(1 - (1 + x)*y)). (End)

Edited by Franck Maminirina Ramaharo, Dec 22 2018 (previous definition and examples were the same as A168620, but with different entries, as pointed out by R. J. Mathar, Oct 21 2012)

cp's OEIS Frontend

A168620 Table T(n,k) with the coefficient [x^k] of the polynomial 5(x+1)^n - 4(x^n+1) in column 0<=k<=n. T(0,0)=1.

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Mathematica

A154312 Triangle T(n,k), 0<=k<=n, read by rows, given by [0,1/2,-1/2,0,0,0,0,0,0,0,...] DELTA [2,-1/2,-1/2,2,0,0,0,0,0,0,0 ...] where DELTA is the operator defined in A084938 .

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Formula

A168621 Triangle read by rows: T(n,0) = T(n,n) = 1 for n >= 0, T(n,k) = ((n - 1)! + 1)*binomial(n, k) for 1 <= k <= n - 1, n >= 2.

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cp's OEIS Frontend

A168620 Table T(n,k) with the coefficient [x^k] of the polynomial 5*(x+1)^n - 4*(x^n+1) in column 0<=k<=n. T(0,0)=1.

Views

Author

Keywords

Comments

Examples

Programs

Mathematica

A154312 Triangle T(n,k), 0<=k<=n, read by rows, given by [0,1/2,-1/2,0,0,0,0,0,0,0,...] DELTA [2,-1/2,-1/2,2,0,0,0,0,0,0,0 ...] where DELTA is the operator defined in A084938 .

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A168621 Triangle read by rows: T(n,0) = T(n,n) = 1 for n >= 0, T(n,k) = ((n - 1)! + 1)*binomial(n, k) for 1 <= k <= n - 1, n >= 2.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Maxima

Formula

Extensions

A168620 Table T(n,k) with the coefficient [x^k] of the polynomial 5(x+1)^n - 4(x^n+1) in column 0<=k<=n. T(0,0)=1.