A155863 -id:A155863 - cp's OEIS frontend

A155871 Subtraction of polynomial coefficients of MacMahon A060187 from third derivative of Pascal's triangle A155863: p(x,n)=(If[n == 0, 1, x^n + 1 + xD[( x + 1)^(n + 1), {x, 3}]] - 2^n(1 - x)^(1 + n)*LerchPhi[x, -n, 1/2])/x.

1, 1, -16, -110, -16, -117, -1322, -1322, -117, -512, -9703, -22288, -9703, -512, -1843, -58977, -256363, -256363, -58977, -1843, -6048, -328588, -2477728, -4664934, -2477728, -328588, -6048, -18953, -1751300, -21692852, -69388094

Offset: 3

Roger L. Bagula, Jan 29 2009

Row sums are:
{2, -142, -2878, -42718, -634366, -10289662, -185702398, -3715637758,
-81748930558, -1961988796414, -51011749920766}

			{1, 1},
{-16, -110, -16},
{-117, -1322, -1322, -117},
{-512, -9703, -22288, -9703, -512},
{-1843, -58977, -256363, -256363, -58977, -1843},
{-6048, -328588, -2477728, -4664934, -2477728, -328588, -6048},
{-18953, -1751300, -21692852, -69388094, -69388094, -21692852, -1751300, -18953},
{-58048, -9108221, -178273184, -906867842, -1527023168, -906867842, -178273184, -9108221, -58048},
{-175815, -46690547, -1403033205, -10836712218, -28587853494, -28587853494, -10836712218, -1403033205, -46690547, -175815},
{-529712, -237214810, -10708833968, -121383574287, -477020204064, -743288082732, -477020204064, -121383574287, -10708833968, -237214810, -529712},
{-1592125, -1198358670, -79944129566, -1295922974075, -7310749751463, -16818058154484, -16818058154484, -7310749751463, -1295922974075, -79944129566, -1198358670, -1592125}

A060187, A155863

p[x_, n_] = (If[n == 0, 1, x^n + 1 + x*D[(x + 1)^(n + 1), {x, 3}]] - 2^n*(1 - x)^(1 + n)*LerchPhi[x, -n, 1/2])/x;
Table[FullSimplify[ExpandAll[p[x, n]]], {n, 3, 13}];
a = Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 3, 13}];
Flatten[a]

p(x,n)=(If[n == 0, 1, x^n + 1 + x*D[( x + 1)^(n + 1), {x, 3}]]
- 2^n*(1 - x)^(1 + n)*LerchPhi[x, -n, 1/2])/x;
t(n,m)=coefficients(p(x,n))

A155865 Triangle T(n,k) = (n-1)*binomial(n-2, k-1) for 1 <= k <= n-1, n >= 2, and T(n,0) = T(n,n) = 1 for n >= 0, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 6, 3, 1, 1, 4, 12, 12, 4, 1, 1, 5, 20, 30, 20, 5, 1, 1, 6, 30, 60, 60, 30, 6, 1, 1, 7, 42, 105, 140, 105, 42, 7, 1, 1, 8, 56, 168, 280, 280, 168, 56, 8, 1, 1, 9, 72, 252, 504, 630, 504, 252, 72, 9, 1, 1, 10, 90, 360, 840, 1260, 1260, 840, 360, 90, 10, 1

Offset: 0

Roger L. Bagula, Jan 29 2009

			Triangle begins:
  1;
  1, 1;
  1, 1,  1;
  1, 2,  2,   1;
  1, 3,  6,   3,   1;
  1, 4, 12,  12,   4,   1;
  1, 5, 20,  30,  20,   5,   1;
  1, 6, 30,  60,  60,  30,   6,   1;
  1, 7, 42, 105, 140, 105,  42,   7,  1;
  1, 8, 56, 168, 280, 280, 168,  56,  8, 1;
  1, 9, 72, 252, 504, 630, 504, 252, 72, 9, 1;
  ...
ConvOffs transform of (1, 1, 2, 3) = integers of row 4: (1, 3, 6, 3, 1). _Gary W. Adamson_, Jul 09 2012

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A002457 (T(2*n, n)), A155863, A155864.

A155865:= func< n,k | k eq 0 or k eq n select 1 else (n-1)*Binomial(n-2, k-1) >;
[A155865(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 04 2021

p[x_, n_] = If[n==0, 1, 1 + x^n + x*D[(x+1)^(n-1), {x, 1}]];
Flatten[Table[CoefficientList[ExpandAll[p[x, n]], x], {n, 0, 10}]]
(* or *)
q = 1;
c[n_, q_]= If[n<2, 1, Product[(i-1)^q, {i, 2, n}]];
T[n_, m_, q_]= c[n, q]/(c[m, q]*c[n-m, q]);
Flatten[Table[T[n, m, q], {n,0,12}, {m, 0, n}]] (* Roger L. Bagula, Mar 09 2010 *)

T(n, k) := if k = 0 or k = n then 1 else (n-1)*binomial(n-2, k-1)$ create_list(T(n, k), n, 0, 12, k, 0, n); /* Franck Maminirina Ramaharo, Dec 05 2018 */

def A155865(n,k): return 1 if (k==0 or k==n) else (n-1)*binomial(n-2, k-1)
flatten([[A155865(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 04 2021

T(n, k) = coefficients of (p(n, x)), where p(n, x) = 1 + x^n + x*((d/dx) (x+1)^n) and T(0, 0) = 1.
Define c(n) = Product_{i=2..n} (i - 1), with c(0) = c(1) = 1. Then T(n,m) = c(n)/(c(m)*c(n-m)). - Roger L. Bagula, Mar 09 2010
The triangle is the ConvOffsStoT transform of the natural numbers prefaced with a 1. A row with n integers is the ConvOffs transform of a finite series of the first (n-1) terms in (1, 1, 2, 3, 4, ...).  See A214281 for definitions of the transform. - Gary W. Adamson, Jul 09 2012
Sum_{k=0..n} T(n, k) = 2 + A001787(n-1) - (3/4)*[n==0]. - R. J. Mathar, Jul 17 2012
From Franck Maminirina Ramaharo, Dec 05 2018: (Start)
T(n, k) = (n-1)*binomial(n-2, k-1) with T(n, 0) = T(n, n) = 1.
n-th row polynomial is (1/2)*(1 + (-1)^(2^n) + 2*x^n + (1 + (-1)^(2^n))*(n - 1)*x*(x + 1)^(n - 2)).
G.f.: 1/(1 - y) + 1/(1 - x*y) + x*y^2/(1 - (1 + x)*y)^2 - 1.
E.g.f.: exp(y) + exp(x*y) + x*(1 - (1 - (1 + x)*y)*exp((1 + x)*y))/(1 + x)^2  - 1. (End)
T(2*n, n) = A002457(n). - Alois P. Heinz, Dec 05 2018

Edited and name clarified by Franck Maminirina Ramaharo, Dec 04 2018

A155864 Triangle T(n,k) = n(n-1)binomial(n-2, k-1) for 1 <= k <= n-1, n >= 2, and T(n,0) = T(n,n) = 1 for n >= 0, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 6, 6, 1, 1, 12, 24, 12, 1, 1, 20, 60, 60, 20, 1, 1, 30, 120, 180, 120, 30, 1, 1, 42, 210, 420, 420, 210, 42, 1, 1, 56, 336, 840, 1120, 840, 336, 56, 1, 1, 72, 504, 1512, 2520, 2520, 1512, 504, 72, 1, 1, 90, 720, 2520, 5040, 6300, 5040, 2520, 720, 90, 1

Offset: 0

Roger L. Bagula, Jan 29 2009

			Triangle begins:
  1;
  1,  1;
  1,  2,   1;
  1,  6,   6,    1;
  1, 12,  24,   12,    1;
  1, 20,  60,   60,   20,    1;
  1, 30, 120,  180,  120,   30,    1;
  1, 42, 210,  420,  420,  210,   42,    1;
  1, 56, 336,  840, 1120,  840,  336,   56,   1;
  1, 72, 504, 1512, 2520, 2520, 1512,  504,  72,  1;
  1, 90, 720, 2520, 5040, 6300, 5040, 2520, 720, 90, 1;
  ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A001815, A155863, A155865.

A155864:= func< n,k | k eq 0 or k eq n select 1 else n*(n-1)*Binomial(n-2, k-1) >;
[A155864(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 04 2021

(* First program *)
p[n_, x_]:= p[n,x]= If[n==0, 1, 1 + x^n + x*D[(x+1)^(n), {x, 2}]];
Flatten[Table[CoefficientList[p[n,x], x], {n, 0, 12}]]
(* Second program *)
Table[If[k==0 || k==n, 1, 2*Binomial[n, 2]*Binomial[n-2, k-1]], {n,0,12}, {k,0,n}] //Flatten (* G. C. Greubel, Jun 04 2021 *)

T(n, k) := ratcoef(x^n + n*(n-1)*x*(x+1)^(n-2) + (1 + (-1)^(2^n))/2, x, k)$ create_list(T(n, k), n, 0, 12, k, 0, n); /* Franck Maminirina Ramaharo, Dec 04 2018 */

def A155864(n,k): return 1 if (k==0 or k==n) else n*(n-1)*binomial(n-2,k-1)
flatten([[A155864(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 04 2021

T(n, k) = coefficients of p(n, x), where p(n, x) = 1 + x^n + x*((d/dx)^2 (1+x)^n), with T(0, 0) = 1.
From Franck Maminirina Ramaharo, Dec 04 2018: (Start)
T(n, k) = n*(n-1)*binomial(n-2, k-1) with T(n, 0) = T(n, n) = 1.
n-th row polynomial is x^n + n*(n - 1)*x*(x + 1)^(n - 2) + (1 + (-1)^(2^n))/2.
G.f.: 1/(1 - y) + 1/(1 - x*y) + 2*x*y^2/(1 - y - x*y)^3 - 1.
E.g.f.: exp(y) + exp(x*y) + x*y^2*exp(y + x*y) - 1. (End)
Sum_{k=0..n} T(n, k) = 2 - [n=0] + A001815(n). - G. C. Greubel, Jun 04 2021

Edited and name clarified by Franck Maminirina Ramaharo, Dec 04 2018

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

A155871 Subtraction of polynomial coefficients of MacMahon A060187 from third derivative of Pascal's triangle A155863: p(x,n)=(If[n == 0, 1, x^n + 1 + x*D[( x + 1)^(n + 1), {x, 3}]] - 2^n*(1 - x)^(1 + n)*LerchPhi[x, -n, 1/2])/x.

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A155865 Triangle T(n,k) = (n-1)*binomial(n-2, k-1) for 1 <= k <= n-1, n >= 2, and T(n,0) = T(n,n) = 1 for n >= 0, read by rows.

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A155864 Triangle T(n,k) = n*(n-1)*binomial(n-2, k-1) for 1 <= k <= n-1, n >= 2, and T(n,0) = T(n,n) = 1 for n >= 0, read by rows.

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Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Maxima

Sage

Formula

Extensions

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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Comments

Examples

Crossrefs

Programs

Mathematica

Maxima

Formula

Extensions

A155871 Subtraction of polynomial coefficients of MacMahon A060187 from third derivative of Pascal's triangle A155863: p(x,n)=(If[n == 0, 1, x^n + 1 + xD[( x + 1)^(n + 1), {x, 3}]] - 2^n(1 - x)^(1 + n)*LerchPhi[x, -n, 1/2])/x.

A155864 Triangle T(n,k) = n(n-1)binomial(n-2, k-1) for 1 <= k <= n-1, n >= 2, and T(n,0) = T(n,n) = 1 for n >= 0, read by rows.