A292798 -id:A292798 - cp's OEIS frontend

A103209 Square array T(n,d) read by antidiagonals: number of structurally-different guillotine partitions of a d-dimensional box in R^d by n hyperplanes.

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

Offset: 1

Ralf Stephan, Jan 27 2005

The columns are the row sums of the inverses of the Riordan arrays ((1-d*x)/(1-x),x(1-d*x)/(1-x)), that is, of the Riordan arrays ((1+x-sqrt(1+2(1-2*d)x+x^2)/(2*d*x),(1+x-sqrt(1+2(1-2*d)x+x^2)/(2*d)). - Paul Barry, May 24 2005

			1,...1,....1,.....1,......1,......1,.......1,.......1,.......1,
1,...2,....3,.....4,......5,......6,.......7,.......8,.......9,
1,...6,...15,....28,.....45,.....66,......91,.....120,.....153,
1,..22,...93,...244,....505,....906,....1477,....2248,....3249,
1,..90,..645,..2380,...6345,..13926,...26845,...47160,...77265,
1,.394,.4791,.24868,..85405,.229326,..522739,.1059976,.1968633,
1,1806,37275,272188,1204245,3956106,10663471,24958200,52546473,

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.
Andrei Asinowski and Toufik Mansour, Separable d-Permutations and Guillotine Partitions, arXiv 0803.3414 [math.CO], 2008; Annals of Combinatorics 14 (1) pp.17-43 Springer, 2010; Abstract
Paul Barry, On the Inverses of a Family of Pascal-Like Matrices Defined by Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.5.6.
Jean Cardinal, Stefan Felsner, Topological drawings of complete bipartite graphs, Journal of Computational Geometry 9.1 (2018), 213-246. Also arXiv:1608.08324 [cs.CG], 2016 (The OEIS is referenced in version v1 but not in v2).

Second column is A006318 (Schroeder numbers), others are A103210 and A103211. Main diagonal is A292798, diagonal under the main diagonal is A103212.

T := (n,k) -> hypergeom([-n, n+1], [2], -k);
seq(print(seq(simplify(T(n, k)), k=0..9)), n=0..6); # Peter Luschny, May 23 2014

T[0, ] = T[, 0] = 1;
T[n_, k_] := Sum[Binomial[n+j, 2j] k^j CatalanNumber[j], {j, 0, n}];
Table[T[n-k+1, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 20 2018, after Paul Barry *)

T(n, d) = (1/n) * sum[i=0..n-1, C(n, i)*C(n, i+1)*(d-1)^i*d^(n-i) ], T(n, 0)=1.
G.f. of d-th column: [1-z-(z^2-4dz+2z+1)^(1/2)]/(2dz-2z).
T(n, k) = sum{j=0..n, C(n+j, 2j)*k^j*C(j)}, C(n) as in A000108. - Paul Barry, May 21 2005
T(n, k) = hypergeom([-n, n+1], [2], -k). - Peter Luschny, May 23 2014

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.

Original entry on oeis.org

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

A302286 a(n) = [x^n] 1/(1 - nx - x/(1 - nx - x/(1 - nx - x/(1 - nx - x/(1 - ...))))), a continued fraction.

Original entry on oeis.org

1, 2, 12, 116, 1530, 25422, 507696, 11814728, 313426350, 9324499610, 307171539576, 11091813369276, 435408606414964, 18453269887229478, 839464708754178240, 40786587211854543120, 2107367668847505288726, 115352793604678609311282, 6667002839420189781109800, 405656528458830256952396420

Offset: 0

Ilya Gutkovskiy, Apr 04 2018

nonn

Main diagonal of A247507.

Cf. A006318, A099169, A247496, A292798.

Table[SeriesCoefficient[1/(1 - n x + ContinuedFractionK[-x, 1 - n x, {k, 1, n}]), {x, 0, n}], {n, 0, 19}]
Table[SeriesCoefficient[(1 - n x - Sqrt[1 - (2 n + 4) x + n^2 x^2])/(2 x), {x, 0, n}], {n, 0, 19}]
Join[{1}, Table[(1/n) Sum[(n + 1)^k Binomial[n, k] Binomial[n, k - 1], {k, 0, n}], {n, 1, 19}]]
Table[(n + 1) Hypergeometric2F1[1 - n, -n, 2, n + 1], {n, 0, 19}]

a(n) = [x^n] (1 - n*x - sqrt(1 - (2*n + 4)*x + n^2*x^2))/(2*x).
a(0) = 1; a(n) = (1/n)*Sum_{k=0..n} (n + 1)^k*binomial(n,k)*binomial(n,k-1).
a(n) = A247507(n,n).
a(n) ~ exp(2*sqrt(n)) * n^(n - 3/4) / (2*sqrt(Pi)). - Vaclav Kotesovec, Jun 08 2019

A366038 a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(n+k,k) * binomial(n(n+1),n-k) n^k.

Original entry on oeis.org

1, 2, 25, 658, 27193, 1548526, 112916830, 10062563610, 1061196371665, 129369938790070, 17909387604206371, 2776290021986848588, 476539253976442601735, 89736215305419802692184, 18395742890606906720656524, 4078527943680251523126851306, 972490249766494185823234587681

Offset: 0

Ilya Gutkovskiy, Sep 26 2023

nonn

Cf. A006318, A064063, A135861, A291699, A292798, A365816, A366012, A366016, A366037.

A366038 := proc(n)
    add(binomial(n+k,k)*binomial(n*(n+1),n-k)*n^k,k=0..n) ;
    %/(n+1) ;
end proc:
seq(A366038(n),n=0..80) ; # R. J. Mathar, Oct 24 2024

Unprotect[Power]; 0^0 = 1; Table[1/(n + 1) Sum[Binomial[n + k, k] Binomial[n (n + 1) , n - k] n^k, {k, 0, n}], {n, 0, 16}]
Table[Binomial[n (n + 1), n] Hypergeometric2F1[-n, n + 1, n^2 + 1, -n]/(n + 1), {n, 0, 16}]
Table[SeriesCoefficient[(1/x) InverseSeries[Series[x (1 - n x)/(1 + x)^n, {x, 0, n + 1}], x], {x, 0, n}], {n, 0, 16}]

a(n) = [x^n] (1/x) * Series_Reversion( x * (1 - n * x) / (1 + x)^n ).

a(n) ~ phi^(3*n + 3/2) * exp(n/phi^2 + 1/(2*phi)) * n^(n - 3/2) / (5^(1/4) * sqrt(2*Pi)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Sep 27 2023

A304936 a(n) = [x^n] 1/(1 - nx/(1 - x - nx/(1 - x - nx/(1 - x - nx/(1 - x - n*x/(1 - ...)))))), a continued fraction.

Original entry on oeis.org

1, 1, 10, 183, 5076, 191105, 9140118, 531731935, 36496595656, 2889768574449, 259443165181410, 26054614893427703, 2894791106297891100, 352618782117325104849, 46736101530152250554926, 6696645353339606889836415, 1031600569146491935984293648, 170029083604373881344301895585

Offset: 0

Ilya Gutkovskiy, May 21 2018

nonn

Cf. A001003, A107841, A131763, A131765, A131846, A131869, A131926, A131927, A292798.

Table[SeriesCoefficient[1/(1 + ContinuedFractionK[-n x, 1 - x, {i, 1, n}]), {x, 0, n}], {n, 0, 17}]
Table[SeriesCoefficient[2/(1 + x + Sqrt[1 - x (2 + 4 n - x)]), {x, 0, n}], {n, 0, 17}]
Table[Sum[(-1)^(n - k) (n + 1)^k Binomial[n, k] Binomial[n + k, k]/(k + 1),{k, 0, n}], {n, 0, 17}]
Table[(-1)^n Hypergeometric2F1[-n, n + 1, 2, n + 1], {n, 0, 17}]

a(n) = [x^n] 2/(1 + x + sqrt(1 - x*(2 + 4*n - x))).
a(n) = Sum_{k=0..n} (-1)^(n-k)*(n + 1)^k*binomial(n,k)*binomial(n+k,k)/(k + 1).
a(n) ~ exp(1/2) * 2^(2*n) * n^(n - 3/2) / sqrt(Pi). - Vaclav Kotesovec, Jun 08 2019

cp's OEIS Frontend

A103209 Square array T(n,d) read by antidiagonals: number of structurally-different guillotine partitions of a d-dimensional box in R^d by n hyperplanes.

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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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A302286 a(n) = [x^n] 1/(1 - n*x - x/(1 - n*x - x/(1 - n*x - x/(1 - n*x - x/(1 - ...))))), a continued fraction.

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A366038 a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(n+k,k) * binomial(n*(n+1),n-k) * n^k.

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A304936 a(n) = [x^n] 1/(1 - n*x/(1 - x - n*x/(1 - x - n*x/(1 - x - n*x/(1 - x - n*x/(1 - ...)))))), a continued fraction.

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Crossrefs

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Mathematica

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.

A302286 a(n) = [x^n] 1/(1 - nx - x/(1 - nx - x/(1 - nx - x/(1 - nx - x/(1 - ...))))), a continued fraction.

A366038 a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(n+k,k) * binomial(n(n+1),n-k) n^k.

A304936 a(n) = [x^n] 1/(1 - nx/(1 - x - nx/(1 - x - nx/(1 - x - nx/(1 - x - n*x/(1 - ...)))))), a continued fraction.