A103209 -id:A103209 - cp's OEIS frontend

A103212 a(n) = (1/n) * Sum_{i=0..n-1} C(n,i)C(n,i+1)(n-1)^i*n^(n-i) for n>=1, a(0)=1.

1, 1, 6, 93, 2380, 85405, 3956106, 224939113, 15175702200, 1185580310121, 105302043709390, 10482085765658661, 1156062800841590148, 139945327558704629221, 18449221488652046992914, 2631255715262150125502865, 403689862107153669227378416, 66297391981691913179574751633

Offset: 0

Ralf Stephan, Jan 27 2005

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A103209, A292798.

Table[HypergeometricPFQ[{-n, n+1}, {2}, -n+1], {n, 0, 20}] (* Vaclav Kotesovec, Sep 24 2017 *)
Flatten[{1, 1, Table[Sum[Binomial[n, k]*Binomial[n, k+1]*(n-1)^k*n^(n-k), {k, 0, n-1}]/n, {n, 2, 20}]}] (* Vaclav Kotesovec, Sep 24 2017 *)

a(n) = {if(n==0, 1, sum(i=0, n-1, binomial(n,i)*binomial(n,i+1)*(n-1)^i*n^(n-i))/n)} \\ Andrew Howroyd, Apr 14 2021

a(n) = A103209(n, n-1). [corrected by Vaclav Kotesovec, Sep 24 2017]

a(n) ~ 2^(2*n) * n^(n-3/2) / (sqrt(Pi) * exp(1/2)). - Vaclav Kotesovec, Sep 24 2017

Prepended a(0)=1 from Vaclav Kotesovec, Sep 24 2017

Terms a(15) and beyond from Andrew Howroyd, Apr 14 2021

A107702 Triangle related to guillotine partitions of a k-dimensional box by n hyperplanes.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 6, 1, 1, 4, 15, 22, 1, 1, 5, 28, 93, 90, 1, 1, 6, 45, 244, 645, 394, 1, 1, 7, 66, 505, 2380, 4791, 1806, 1, 1, 8, 91, 906, 6345, 24868, 37275, 8558, 1, 1, 9, 120, 1477, 13926, 85405, 272188, 299865, 41586, 1, 1, 10, 153, 2248, 26845, 229326, 1204245, 3080596, 2474025, 206098, 1

Offset: 0

Paul Barry, May 21 2005

Row sums are A107703. Transpose of square array A103209, read by antidiagonals.

			Triangle begins:
  1;
  1, 1;
  1, 2,  1;
  1, 3,  6,   1;
  1, 4, 15,  22,    1;
  1, 5, 28,  93,   90,     1;
  1, 6, 45, 244,  645,   394,     1;
  1, 7, 66, 505, 2380,  4791,  1806,    1;
  1, 8, 91, 906, 6345, 24868, 37275, 8558, 1;
  ...

Seiichi Manyama, Rows n = 0..139, flattened
E. Ackerman, G. Barequet, R. Y. Pinter and D. Romik, The number of guillotine partitions in d dimensions, Inf. Proc. Lett 98 (4) (2006) 162-167.

Diagonals: A000012, A006318, A103210, A103211, A133305, A133306, A133307, A133308, A133309. - Philippe Deléham, Dec 10 2008

Cf. A000384, A103209.

T(n, k) = sum(j=0, k, (n-k)^j*binomial(k+j, 2*j)*binomial(2*j, j)/(j+1)); \\ Seiichi Manyama, Oct 02 2023

Number triangle T(n, k)=if(k<=n, sum{j=0..k, C(k+j, 2j)(n-k)^j*C(j)}, 0), C(n) given by A000108.

A297899 Triangle read by rows, T(n, k) = binomial(n, k)*hypergeom([k-n, n+1], [k+2], -4), for n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 5, 1, 45, 10, 1, 505, 115, 15, 1, 6345, 1460, 210, 20, 1, 85405, 19765, 2990, 330, 25, 1, 1204245, 279710, 43635, 5220, 475, 30, 1, 17558705, 4088615, 651165, 81955, 8275, 645, 35, 1, 262577745, 61254760, 9901860, 1290520, 139350, 12280, 840, 40, 1

Offset: 0

Peter Luschny, Jan 08 2018

			Triangle starts:
[0]       1
[1]       5,      1
[2]      45,     10,     1
[3]     505,    115,    15,    1
[4]    6345,   1460,   210,   20,   1
[5]   85405,  19765,  2990,  330,  25,  1
[6] 1204245, 279710, 43635, 5220, 475, 30, 1

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

T(n, 0) = A133305(n). Row sums are A297705, alternating row sums are A131765.

Cf. A103209.

T[n_, k_] := Binomial[n, k] Hypergeometric2F1[k - n, n + 1, k + 2, -4];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten
T[n_, k_] := Sum[4^(j - k)*(k + 1)*Binomial[n + j - k, 2*j - k]*Binomial[2*j - k, j - k]/(j + 1), {j, k, n}];
Flatten[Table[T[n, k], {n, 0, 8}, {k, 0, n}]] (* Detlef Meya, Jan 15 2024 *)

T(n,k) = sum(j = k, n, 4^(j - k)*(k + 1)*binomial(n + j - k, 2*j - k)* binomial(2*j - k, j - k)/(j + 1)) \\ Andrew Howroyd, Jan 15 2024

T(n, k) = Sum_{j = k..n} 4^(j - k)*(k + 1)*binomial(n + j - k, 2*j - k)* binomial(2*j - k, j - k)/(j + 1). - Detlef Meya, Jan 15 2024

cp's OEIS Frontend

A103212 a(n) = (1/n) * Sum_{i=0..n-1} C(n,i)C(n,i+1)(n-1)^i*n^(n-i) for n>=1, a(0)=1.

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A107702 Triangle related to guillotine partitions of a k-dimensional box by n hyperplanes.

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PARI

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A297899 Triangle read by rows, T(n, k) = binomial(n, k)*hypergeom([k-n, n+1], [k+2], -4), for n >= 0 and 0 <= k <= n.

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cp's OEIS Frontend

A103212 a(n) = (1/n) * Sum_{i=0..n-1} C(n,i)*C(n,i+1)*(n-1)^i*n^(n-i) for n>=1, a(0)=1.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A107702 Triangle related to guillotine partitions of a k-dimensional box by n hyperplanes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A297899 Triangle read by rows, T(n, k) = binomial(n, k)*hypergeom([k-n, n+1], [k+2], -4), for n >= 0 and 0 <= k <= n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A103212 a(n) = (1/n) * Sum_{i=0..n-1} C(n,i)C(n,i+1)(n-1)^i*n^(n-i) for n>=1, a(0)=1.