A103211 -id:A103211 - cp's OEIS frontend

A133309 a(n) = (1/n)Sum_{i=0..n-1} C(n,i)C(n,i+1)8^i9^(n-i), a(0)=1.

1, 9, 153, 3249, 77265, 1968633, 52546473, 1450365921, 41058670113, 1185580310121, 34783088255289, 1033907690362257, 31070005849929969, 942384250116160857, 28812102048874578249, 887007207177728561601, 27473495809057571051073, 855518113376312857290441

Offset: 0

Philippe Deléham, Oct 18 2007

nonn

Ninth column of array A103209.

The Hankel transform of this sequence is 72^C(n+1,2). - Philippe Deléham, Oct 29 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000108, A060693, A103209, A103210, A103211.

Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!( (1-x-Sqrt(x^2-34*x+1))/16 )); // G. C. Greubel, Feb 10 2018

Rest@ CoefficientList[ Series[(1-x-Sqrt[x^2-34*x+1])/16, {x, 0, 18}], x] (* Robert G. Wilson v, Oct 19 2007 *)
Table[-((3 I LegendreP[n, -1, 2, 17])/(2 Sqrt[2])), {n, 0, 20}] (* Vaclav Kotesovec, Aug 13 2013 *)

my(x='x+O('x^30)); Vec((1-x-sqrt(x^2-34*x+1))/16) \\ G. C. Greubel, Feb 10 2018

G.f.: (1-z-sqrt(z^2-34*z+1))/16.
a(n) = Sum_{k=0..n} A088617(n,k)*8^k.
a(n) = Sum_{k=0..n} A060693(n,k)*8^(n-k).
a(n) = Sum_{k=0..n} C(n+k, 2k)8^k*C(k), C(n) given by A000108.
a(0)=1, a(n) = a(n-1) + 8*Sum_{k=0..n-1} a(k)*a(n-1-k). - Philippe Deléham, Oct 23 2007
a(n) ~ sqrt(144+102*sqrt(2))*(17+12*sqrt(2))^n/(16*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 13 2013
Recurrence: (n+1)*a(n) = 17*(2*n-1)*a(n-1) - (n-2)*a(n-2). - Vaclav Kotesovec, Aug 13 2013
G.f.: 1/(1 - 9*x/(1 - 8*x/(1 - 9*x/(1 - 8*x/(1 - 9*x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, May 10 2017

More terms from Robert G. Wilson v, Oct 19 2007

A152601 a(n) = Sum_{k=0..n} C(n+k,2k)A000108(k)3^k*2^(n-k).

Original entry on oeis.org

1, 5, 40, 395, 4360, 51530, 637840, 8163095, 107140360, 1434252230, 19507077040, 268796321870, 3744480010960, 52647783144980, 746145741252640, 10648007952942095, 152877753577617160, 2206713692628578030

Offset: 0

Paul Barry, Dec 09 2008

Hankel transform is 15^C(n+1,2).

Cf. A103211, A103210.
Cf. A088617, A060693, A090181, A131198, A133336, A086810
Cf. A152600.

a(n) = A152600(n+1)/2.
a(n) = Sum_{k=0..n} A088617(n,k)*3^k*2^(n-k) = Sum_{k=0..n} A060693(n,k)*2^k*3^(n-k). - Philippe Deléham, Dec 10 2008
a(n) = Sum_{k=0..n} A090181(n,k)*5^k*3^(n-k). - Philippe Deléham, Dec 10 2008
a(n) = Sum_{k=0..n} A131198(n,k)*3^k*5^(n-k). - Philippe Deléham, Dec 10 2008
a(n) = Sum_{k=0..n} A133336(n,k)*(-2)^k*5^(n-k) = Sum_{k=0..n} A086810(n,k)*5^k*(-2)^(n-k). - Philippe Deléham, Dec 10 2008
G.f.: 1/(1-5x/(1-3x/(1-5x/(1-3x/(1-5x/(1-3x/(1-5x/(1-... (continued fraction). - Philippe Deléham, Nov 28 2011
Conjecture: (n+1)*a(n) +8*(-2*n+1)*a(n-1) +4*(n-2)*a(n-2)=0. - R. J. Mathar, Nov 24 2012
G.f.: 1/G(x), with G(x) = 1-2*x-(3*x)/G(x) (continued fraction). - Nikolaos Pantelidis, Jan 09 2023

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.

A349516 G.f. A(x) satisfies: A(x) = (1 + 3 * x * A(x)^3) / (1 - x).

Original entry on oeis.org

1, 4, 40, 544, 8512, 144448, 2584960, 48026368, 917535232, 17911696384, 355725727744, 7164414312448, 145983839272960, 3003998986682368, 62337412584669184, 1303045468017786880, 27411525832634269696, 579884892273731436544, 12328565505725394583552

Offset: 0

Ilya Gutkovskiy, Nov 20 2021

nonn

Cf. A001764, A103211, A346626, A348912, A349517.

nmax = 18; A[] = 0; Do[A[x] = (1 + 3 x A[x]^3)/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = a[n - 1] + 3 Sum[Sum[a[i] a[j] a[n - i - j - 1], {j, 0, n - i - 1}], {i, 0, n - 1}]; Table[a[n], {n, 0, 18}]
Table[Sum[Binomial[n + 2 k, 3 k] Binomial[3 k, k] 3^k/(2 k + 1), {k, 0, n}], {n, 0, 18}]

a(n) = sum(k=0, n, binomial(n+2*k,3*k) * binomial(3*k,k) * 3^k / (2*k+1)) \\ Andrew Howroyd, Nov 20 2021

a(0) = 1; a(n) = a(n-1) + 3 * Sum_{i=0..n-1} Sum_{j=0..n-i-1} a(i) * a(j) * a(n-i-j-1).
a(n) = Sum_{k=0..n} binomial(n+2*k,3*k) * binomial(3*k,k) * 3^k / (2*k+1).
a(n) ~ sqrt(13 + 7*3^(1/3) + 5*3^(2/3)) / (12 * sqrt(Pi) * n^(3/2) * (1 + 3^(4/3)/2 - 3^(5/3)/2)^n). - Vaclav Kotesovec, Nov 21 2021

cp's OEIS Frontend

A133309 a(n) = (1/n)*Sum_{i=0..n-1} C(n,i)*C(n,i+1)*8^i*9^(n-i), a(0)=1.

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A152601 a(n) = Sum_{k=0..n} C(n+k,2k)*A000108(k)*3^k*2^(n-k).

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A349516 G.f. A(x) satisfies: A(x) = (1 + 3 * x * A(x)^3) / (1 - x).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A133309 a(n) = (1/n)Sum_{i=0..n-1} C(n,i)C(n,i+1)8^i9^(n-i), a(0)=1.

A152601 a(n) = Sum_{k=0..n} C(n+k,2k)A000108(k)3^k*2^(n-k).