cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 41-46 of 46 results.

A041005 Convolution of Catalan numbers A000108(n+1), n >= 0, with A020918.

Original entry on oeis.org

1, 16, 159, 1260, 8722, 55152, 326811, 1844084, 10015566, 52754624, 270976342, 1362986520, 6734927460, 32775704608, 157408497171, 747269225028, 3511471892470, 16351481223840, 75525932249922, 346305571781224
Offset: 0

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Author

Wolfdieter Lang

Keywords

Comments

Also convolution of A001791(n+1), n >= 0, with A038845; also convolution of A008549(n+1), n >= 0, with A002802; also convolution of A029760 with A002697; also convolution of A038806(n+1), n >= 0, with A002457; also convolution of A038836 with A000302 (powers of 4); also convolution of A041001 with A000984 (central binomial coefficients).

Formula

a(n)=binomial(n+7, 3)*binomial(2*(n+4), n+2)/20 - (n+4)*(n+3)*4^(n+1); G.f. (c(x)^2)/(1-4*x)^(7/2), where c(x) = g.f. for Catalan numbers.

A118445 Number of tree-rooted maps of genus 1 with n edges: rooted maps on the torus with a distinguished spanning tree.

Original entry on oeis.org

1, 25, 490, 8820, 152460, 2576574, 42942900, 709171320, 11636856660, 190068658780, 3093732938296, 50222937310000, 813611584422000, 13158602740363500, 212528020730913000, 3428785401125396400, 55266606794455402500, 890117467077758188500
Offset: 2

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Author

Valery A. Liskovets, May 04 2006

Keywords

Comments

Tree-rooted planar maps are counted by A005568 and tree-rooted maps of (orientable) genus 2 by A118446. Typically, a(11) = 190068658780 = 2^2*5*7^2*11*13^2*17^2*19^2.

Links

Programs

  • Mathematica
    HypergeometricPFQ[{5/2, 5/2}, {4}, 16x] + O[x]^18 // CoefficientList[#, x]& (* Jean-François Alcover, Aug 28 2019 *)
Table[n*(n-1) * Binomial[2*n,n]^2 / (24*(n+1)), {n, 2, 20}] (* Vaclav Kotesovec, Feb 17 2024 *)

Formula

a(n) = binomial(2n, 0) C(0) b(n) + binomial(2n, 2) C(1) b(n-1) + binomial(2n, 4) C(2) b(n-2) + ... + binomial(2n, 2n) C(n) b(0), where C(n) = A000108(n) - n-th Catalan number and b(n) = (2n-1)!/(6(n-2)! (n-1)!) = A002802(n-2) - the number of toroidal one-vertex maps with n edges for n >= 2 and b(0) = b(1) = 0.
O.g.f.: x^2 * hypergeom([5/2, 5/2], [4], 16*x). - Mark van Hoeij, Apr 06 2013
D-finite with recurrence -(n+1)*(n-2)*a(n) +4*((2*n-1)^2)*a(n-1)=0. - R. J. Mathar, Jul 27 2022
From Vaclav Kotesovec, Feb 17 2024: (Start)
a(n) = n*(n-1) * binomial(2*n,n)^2 / (24*(n+1)).
a(n) ~ 2^(4*n-3)/(3*Pi). (End)

Extensions

Added more terms, Joerg Arndt, Apr 07 2013

A370237 Number of genus 3 partitions of the n-set.

Original entry on oeis.org

1, 94, 2620, 45430, 600655, 6633484, 64336844, 565256120
Offset: 8

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Author

Robert Coquereaux, Feb 12 2024

Keywords

Comments

Call B(n, g) the number of genus g partitions of a set with n elements (genus-dependent Bell number). Then a(n) = B(n, 3) with B(8, 3) = 1.
a(8) = 1 through a(15) = 565256120 were explicitly determined by listing of partitions of an n-set and selecting those of genus 3.
The coefficients of the sixth-degree polynomial appearing in the numerator of the conjectured formula were determined by using experimental values for a(8) up to a(14); the term a(15) given by the formula agrees with the experimental value.
Using the conjectured formula for a(n) gives the following terms for n=16..20 : 4593034160, 35025118700, 253374008888, 1753071498620, 11675101781850. The E.g.f. given in the Formula section is obtained from the conjectured formula for a(n).

Links

Crossrefs

Column g=3 of A370235.
Cf. A000108, A002802, A297179, A001263, A370236, A297178.

Formula

Conjecture: a(n) = (1/(2^13 * 3^4 * 5 * 7)) * (35*n^6 - 819*n^5 + 7589*n^4 - 36009*n^3 + 93464*n^2 - 129060*n + 95040)/((2*n - 11)*(2*n - 9)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)) * (1/(n-8)!) * (2*n)!/n!.
Conjecture: E.g.f.: (1/181440)*exp(2*x)*(x^2*(720 - 720*x + 1080*x^2 - 720*x^3 + 537*x^4 - 294*x^5 + 140*x^6)*BesselI(0, 2*x) + x*(-720 + 720*x - 1440*x^2 + 1080*x^3 - 1017*x^4 + 594*x^5 - 329*x^6 + 140*x^7)*BesselI(1, 2*x)).

A382274 Expansion of 1/(1 - 4*x/(1-x)^2)^(5/2).

Original entry on oeis.org

1, 10, 90, 730, 5570, 40762, 289370, 2007210, 13671170, 91750250, 608294490, 3991833210, 25968131010, 167664187290, 1075453670490, 6858654320970, 43517809896450, 274862176368330, 1728960219827290, 10835520927931930, 67679638209628098, 421442759107879930
Offset: 0

Views

Author

Seiichi Manyama, Mar 29 2025

Keywords

Crossrefs

Cf. A110170, A377198, A382332.
Cf. A002802, A377199.

Programs

  • PARI
    a(n) = sum(k=0, n, (-4)^k*binomial(-5/2, k)*binomial(n+k-1, n-k));

Formula

a(0) = 1; a(n) = 2 * Sum_{k=0..n-1} (5-3*k/n) * (n-k) * a(k).
a(n) = ((7*n+3)*a(n-1) - (7*n-24)*a(n-2) + (n-3)*a(n-3))/n for n > 2.
a(n) = Sum_{k=0..n} (-4)^k * binomial(-5/2,k) * binomial(n+k-1,n-k).
a(n) = 10*n*hypergeom([7/2, 1-n, 1+n], [3/2, 2], -1) for n > 0. - Stefano Spezia, Mar 30 2025
a(n) ~ 2^(3/4) * n^(3/2) * (1 + sqrt(2))^(2*n) / (3*sqrt(Pi)). - Vaclav Kotesovec, Apr 13 2025

A090299 Table T(n,k), n>=0 and k>=0, read by antidiagonals : the k-th column given by the k-th polynomial K_k related to A090285.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 5, 10, 5, 1, 14, 35, 22, 7, 1, 42, 126, 93, 38, 9, 1, 132, 462, 386, 187, 58, 11, 1, 429, 1716, 1586, 874, 325, 82, 13, 1, 1430, 6435, 6476, 3958, 1686, 515, 110, 15, 1, 4862, 24310, 26333, 17548, 8330, 2934, 765, 142, 17, 1
Offset: 0

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Author

Philippe Deléham, Jan 25 2004

Keywords

Comments

Read as a number triangle, this is the Riordan array (c(x),x/sqrt(1-4x)) where c(x) is the g.f. of A000108. - Paul Barry, May 16 2005

Examples

			row n=0 : 1, 1, 2, 5, 14, 42, 132, 429, ... see A000108.
row n=1 : 1, 3, 10, 35, 126, 462, 1716, 6435, ... see A001700.
row n=2 : 1, 5, 22, 93, 386, 1586, 6476, ... see A000346.
row n=3 : 1, 7, 38, 187, 874, 3958, 17548, ... see A000531.
row n=4 : 1, 9, 58, 325, 1686, 8330, 39796, ... see A018218.

Crossrefs

Other rows : A029887, A042941, A045724, A042985, A045492. Columns : A000012, A005408. Row n is the convolution of the row (n-j) with A000984, A000302, A002457, A002697 (first term omitted), A002802, A038845, A020918, A038846, A020920 for j=1, 2, ..9 respectively.

Formula

T(n, k) = K_k(n)= Sum_{j>=0} A090285(k, j)*2^j*binomial(n, j). T(n, 1) = 2*n+1. T(n, 2) = 2*A028387(n).

Extensions

Corrected by Alford Arnold, Oct 18 2006

A353596 Triangle read by rows, T(n, k) = [x^k] (-2)^n*GegenbauerC(n, -1/2, x).

Original entry on oeis.org

1, 0, 2, 2, 0, -2, 0, -4, 0, 4, -2, 0, 12, 0, -10, 0, 12, 0, -40, 0, 28, 4, 0, -60, 0, 140, 0, -84, 0, -40, 0, 280, 0, -504, 0, 264, -10, 0, 280, 0, -1260, 0, 1848, 0, -858, 0, 140, 0, -1680, 0, 5544, 0, -6864, 0, 2860, 28, 0, -1260, 0, 9240, 0, -24024, 0, 25740, 0, -9724
Offset: 0

Views

Author

Peter Luschny, May 06 2022

Keywords

Examples

			Triangle T(n, k) starts:
[0]   1;
[1]   0,   2;
[2]   2,   0,  -2;
[3]   0,  -4,   0,     4;
[4]  -2,   0,  12,     0,   -10;
[5]   0,  12,   0,   -40,     0,   28;
[6]   4,   0, -60,     0,   140,    0,  -84;
[7]   0, -40,   0,   280,     0, -504,    0,   264;
[8] -10,   0, 280,     0, -1260,    0, 1848,     0, -858;
[9]   0, 140,   0, -1680,     0, 5544,    0, -6864,    0, 2860;
.
Unsigned antidiagonals |T(n+k, n-k)|:
[0]  1;
[1]  2,   2;
[2]  2,   4,    2;
[3]  4,  12,   12,    4;
[4] 10,  40,   60,   40,   10;
[5] 28, 140,  280,  280,  140,  28;
[6] 84, 504, 1260, 1680, 1260, 504, 84;

Crossrefs

Diagonals (also divided by 2^k): A002420 (main), A028329 (main-2) (also A000984), A005430 (main-4) (also A002457), A002802 (main-6).

Programs

  • Maple
    g := n -> (-2)^n*GegenbauerC(n, -1/2, x):
seq(print(seq(coeff(simplify(g(n)), x, k), k = 0..n)), n = 0..9);
  • Mathematica
    s={}; For[n=0,n<11,n++,For[k=0,kDetlef Meya, Oct 03 2023 *)
Previous Showing 41-46 of 46 results.

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