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.

Showing 1-10 of 24 results. Next

A349310 G.f. A(x) satisfies: A(x) = (1 + x * A(x)^4) / (1 - x).

Original entry on oeis.org

1, 2, 10, 74, 642, 6082, 60970, 635818, 6826690, 74958914, 837833482, 9500939978, 109037364930, 1264049402754, 14780619799722, 174121322204074, 2064572904600706, 24620095821589378, 295087003429677322, 3552841638851183690, 42950428996378731010
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 14 2021

Keywords

Links

Crossrefs

Cf. A002293, A006318, A346626, A346646, A349311, A349312, A349313, A349314.

Programs

  • Mathematica
    nmax = 20; A[] = 0; Do[A[x] = (1 + x A[x]^4)/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n + 3 k, 4 k] Binomial[4 k, k]/(3 k + 1), {k, 0, n}], {n, 0, 20}]

Formula

a(n) = Sum_{k=0..n} binomial(n+3*k,4*k) * binomial(4*k,k) / (3*k+1).
a(n) = F([(1+n)/3, (2+n)/3, (3+n)/3, -n], [2/3, 1, 4/3], -1), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 14 2021
a(n) ~ sqrt(1 + 3*r) / (2^(13/6) * sqrt(3*Pi) * (1-r)^(1/6) * n^(3/2) * r^(n + 1/3)), where r = 0.0766602099042102089064087954661556186872273232742446843... is the smallest real root of the equation 3^3 * (1-r)^4 = 4^4 * r. - Vaclav Kotesovec, Nov 15 2021

A364393 G.f. satisfies A(x) = 1 + x*(1 + 1/A(x)^2).

Original entry on oeis.org

1, 2, -4, 20, -120, 800, -5696, 42416, -326304, 2572992, -20685696, 168920704, -1397257472, 11682707712, -98578346496, 838369268480, -7178912946688, 61842549386240, -535575159363584, 4660216874719232, -40722264390799360, 357204260381327360
Offset: 0

Views

Author

Seiichi Manyama, Jul 22 2023

Keywords

Crossrefs

Cf. A364395, A364397, A364399.
Cf. A346626.

Programs

  • Maple
    A364393 := proc(n)
    if n = 0 then
        1;
    else
        (-1)^(n-1)*add( binomial(n,k) * binomial(n+2*k-2,n-1),k=0..n)/n ;
    end if;
end proc:
seq(A364393(n),n=0..70); # R. J. Mathar, Jul 25 2023
  • Mathematica
    m = 22;
A[_] = 1;
Do[A[x_] = 1 + x*(1 + 1/A[x]^2) + O[x]^m // Normal, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Sep 05 2023 *)
  • PARI
    a(n) = if(n==0, 1, (-1)^(n-1)*sum(k=0, n, binomial(n, k)*binomial(n+2*k-2, n-1))/n);

Formula

G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A346626.
a(n) = (-1)^(n-1) * (1/n) * Sum_{k=0..n} binomial(n,k) * binomial(n+2*k-2,n-1) for n > 0.
D-finite with recurrence 2*n*(2*n-1)*a(n) +(35*n^2-64*n+24) *a(n-1) +(-35*n^2+205*n-288) *a(n-2) +2*(-43*n^2+341*n-660) *a(n-3) -4*(7*n-30)*(n-5) *a(n-4) -8*(n-5)*(n-6)*a(n-5)=0. - R. J. Mathar, Jul 25 2023
a(n) = (-1)^(n-1)*n*3F2([1-n, (n+1)/2, n/2+1], [3/2, 2], -1) for n > 1. - Stefano Spezia, Oct 21 2023

A349311 G.f. A(x) satisfies: A(x) = (1 + x * A(x)^5) / (1 - x).

Original entry on oeis.org

1, 2, 12, 112, 1232, 14832, 189184, 2512064, 34358784, 480745984, 6848734464, 99003237376, 1448575666176, 21411827808256, 319255531155456, 4796005997940736, 72520546008219648, 1102912584949792768, 16859182461720526848, 258886644574700699648
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 14 2021

Keywords

Links

Crossrefs

Cf. A002294, A006318, A346626, A346647, A349310, A349312, A349313, A349314.

Programs

  • Mathematica
    nmax = 19; A[] = 0; Do[A[x] = (1 + x A[x]^5)/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n + 4 k, 5 k] Binomial[5 k, k]/(4 k + 1), {k, 0, n}], {n, 0, 19}]

Formula

a(n) = Sum_{k=0..n} binomial(n+4*k,5*k) * binomial(5*k,k) / (4*k+1).
a(n) = F([(1+n)/4, (2+n)/4, (3+n)/4, (4+n)/4, -n], [1/2, 3/4, 1, 5/4], -1), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 14 2021
a(n) ~ sqrt(1 + 4*r) / (2 * 5^(3/4) * sqrt(2*Pi) * (1-r)^(1/4) * n^(3/2) * r^(n + 1/4)), where r = 0.0600920016324256496641829206872407657377702010870270617... is the real root of the equation 4^4 * (1-r)^5 = 5^5 * r. - Vaclav Kotesovec, Nov 15 2021

A349312 G.f. A(x) satisfies: A(x) = (1 + x * A(x)^6) / (1 - x).

Original entry on oeis.org

1, 2, 14, 158, 2106, 30762, 476406, 7683926, 127692530, 2171184146, 37592376734, 660522703886, 11747865153962, 211093333172282, 3826315983647366, 69880933123237958, 1284661783610775010, 23753502514840942882, 441458929706855144494, 8242097867816771820926
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 14 2021

Keywords

Links

Crossrefs

Cf. A002295, A006318, A346626, A346648, A349310, A349311, A349313, A349314.

Programs

  • Mathematica
    nmax = 19; A[] = 0; Do[A[x] = (1 + x A[x]^6)/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n + 5 k, 6 k] Binomial[6 k, k]/(5 k + 1), {k, 0, n}], {n, 0, 19}]

Formula

a(n) = Sum_{k=0..n} binomial(n+5*k,6*k) * binomial(6*k,k) / (5*k+1).
a(n) = F([(1+n)/5, (2+n)/5, (3+n)/5, (4+n)/5, 1+n/5, -n], [2/5, 3/5, 4/5, 1, 6/5], -1), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 14 2021
a(n) ~ sqrt(1 + 5*r) / (2^(6/5) * 3^(7/10) * sqrt(5*Pi) * (1-r)^(3/10) * n^(3/2) * r^(n + 1/5)), where r = 0.04941755525635041337247049893940451999923592381716... is the smallest real root of the equation 5^5 * (1-r)^6 = 6^6 * r. - Vaclav Kotesovec, Nov 15 2021

A349313 G.f. A(x) satisfies: A(x) = (1 + x * A(x)^7) / (1 - x).

Original entry on oeis.org

1, 2, 16, 212, 3320, 57024, 1038928, 19718512, 385668448, 7718866880, 157326086656, 3254310606208, 68142850580480, 1441588339943168, 30765576147680000, 661561298256228096, 14319744815795062272, 311756656998135770112, 6822215641015820419072
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 14 2021

Keywords

Links

Crossrefs

Cf. A002296, A006318, A346626, A346649, A349310, A349311, A349312, A349314.

Programs

  • Mathematica
    nmax = 18; A[] = 0; Do[A[x] = (1 + x A[x]^7)/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n + 6 k, 7 k] Binomial[7 k, k]/(6 k + 1), {k, 0, n}], {n, 0, 18}]

Formula

a(n) = Sum_{k=0..n} binomial(n+6*k,7*k) * binomial(7*k,k) / (6*k+1).
a(n) = F([(1+n)/6, (2+n)/6, (3+n)/6, (4+n)/6, (5+n)/6, 1+n/6, -n], [1/3, 1/2, 2/3, 5/6, 1, 7/6], -1), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 14 2021
a(n) ~ sqrt(1 + 6*r) / (2 * 7^(2/3) * sqrt(3*Pi) * (1-r)^(1/3) * n^(3/2) * r^(n + 1/6)), where r = 0.04196526794785323647696104132939153750367778616407409162... is the real root of the equation 6^6 * (1-r)^7 = 7^7 * r. - Vaclav Kotesovec, Nov 15 2021

A366363 G.f. satisfies A(x) = (1 + x/A(x))/(1 - x).

Original entry on oeis.org

1, 2, 0, 4, -8, 32, -112, 432, -1696, 6848, -28160, 117632, -497664, 2128128, -9183488, 39940864, -174897664, 770452480, -3411959808, 15181264896, -67833868288, 304256253952, -1369404661760, 6182858317824, -27995941060608, 127100310290432, -578433619525632
Offset: 0

Views

Author

Seiichi Manyama, Oct 08 2023

Keywords

Links

Crossrefs

Cf. A346626, A349310, A349311, A349312, A349313, A349314, A366364, A366365, A366366.
Cf. A366356.

Programs

  • Mathematica
    A366363[n_]:=(-1)^(n-1)Sum[Binomial[2k-1,k]Binomial[k-1,n-k]/(2k-1),{k,0,n}];
Array[A366363,30,0] (* Paolo Xausa, Oct 20 2023 *)
  • PARI
    a(n) = (-1)^(n-1)*sum(k=0, n, binomial(2*k-1, k)*binomial(k-1, n-k)/(2*k-1));

Formula

G.f.: A(x) = -2*x / (1-sqrt(1+4*x*(1-x))).
a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(2*k-1,k) * binomial(k-1,n-k)/(2*k-1).

A366364 G.f. satisfies A(x) = (1 + x/A(x)^2)/(1 - x).

Original entry on oeis.org

1, 2, -2, 14, -70, 426, -2714, 18118, -124814, 881042, -6339058, 46318334, -342769750, 2563781690, -19350683018, 147197511222, -1127334112542, 8685458120226, -67270210217186, 523472089991662, -4090668558473318, 32088204418069450, -252576222775705466
Offset: 0

Views

Author

Seiichi Manyama, Oct 08 2023

Keywords

Crossrefs

Cf. A346626, A349310, A349311, A349312, A349313, A349314, A366363, A366365, A366366.
Cf. A366357.

Programs

  • PARI
    a(n) = (-1)^(n-1)*sum(k=0, n, binomial(3*k-1, k)*binomial(2*k-1, n-k)/(3*k-1));

Formula

a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(3*k-1,k) * binomial(2*k-1,n-k)/(3*k-1).

A349314 G.f. A(x) satisfies: A(x) = (1 + x * A(x)^8) / (1 - x).

Original entry on oeis.org

1, 2, 18, 274, 4930, 97346, 2039570, 44524818, 1001773058, 23065953794, 540886665618, 12872727013522, 310135678438978, 7549240857128258, 185381380643501970, 4586875745951650706, 114244031335228433922, 2862001783406012428802, 72067481493990612275474
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 14 2021

Keywords

Comments

In general, for k > 1, Sum_{j=0..n} binomial(n + (k-1)*j,k*j) * binomial(k*j,j) / ((k-1)*j+1) ~ (1-r)^(1/(k-1) - 1/2) * sqrt(1 + (k-1)*r) / (sqrt(2*Pi*(k-1)) * k^(1/(k-1) + 1/2) * n^(3/2) * r^(n + 1/(k-1))), where r is the smallest real root of the equation (k-1)^(k-1) * (1-r)^k = k^k * r. - Vaclav Kotesovec, Nov 15 2021

Links

Crossrefs

Cf. A006318, A007556, A346626, A346650, A349310, A349311, A349312, A349313.

Programs

  • Mathematica
    nmax = 18; A[] = 0; Do[A[x] = (1 + x A[x]^8)/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n + 7 k, 8 k] Binomial[8 k, k]/(7 k + 1), {k, 0, n}], {n, 0, 18}]

Formula

a(n) = Sum_{k=0..n} binomial(n+7*k,8*k) * binomial(8*k,k) / (7*k+1).
a(n) = F([(1+n)/7, (2+n)/7, (3+n)/7, (4+n)/7, (5+n)/7, (6+n)/7, 1+n/7, -n], [2/7, 3/7, 4/7, 5/7, 6/7, 1, 8/7], -1), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 14 2021
a(n) ~ sqrt(1 + 7*r) / (2^(17/7) * sqrt(7*Pi) * (1-r)^(5/14) * n^(3/2) * r^(n + 1/7)), where r = 0.036466941615119756839260438459647497790132092200414533994... is the smallest real root of the equation 7^7 * (1-r)^8 = 8^8 * r. - Vaclav Kotesovec, Nov 15 2021

A366365 G.f. satisfies A(x) = (1 + x/A(x)^3)/(1 - x).

Original entry on oeis.org

1, 2, -4, 32, -240, 2064, -18816, 179264, -1762816, 17758976, -182342400, 1901196288, -20075427840, 214246524928, -2307200135168, 25039992254464, -273603550461952, 3007387399258112, -33230774508716032, 368915340555517952, -4112806343370539008
Offset: 0

Views

Author

Seiichi Manyama, Oct 08 2023

Keywords

Crossrefs

Cf. A346626, A349310, A349311, A349312, A349313, A349314, A366363, A366364, A366366.
Cf. A366358.

Programs

  • PARI
    a(n) = (-1)^(n-1)*sum(k=0, n, binomial(4*k-1, k)*binomial(3*k-1, n-k)/(4*k-1));

Formula

a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(4*k-1,k) * binomial(3*k-1,n-k)/(4*k-1).

A366184 G.f. A(x) satisfies A(x) = (1 + x*A(x)^3)/(1 - x)^3.

Original entry on oeis.org

1, 4, 21, 163, 1487, 14697, 153226, 1659338, 18483960, 210437161, 2437721418, 28640748192, 340473075541, 4087735789616, 49494986770104, 603699827411356, 7410709463933414, 91484338902961485, 1135029142529785303, 14145212892466682781, 176993823220824229047
Offset: 0

Views

Author

Seiichi Manyama, Oct 03 2023

Keywords

Crossrefs

Cf. A001764, A346626, A364629.
Cf. A364623, A366182, A366183.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(n+8*k+2, n-k)*binomial(3*k, k)/(2*k+1));

Formula

a(n) = Sum_{k=0..n} binomial(n+8*k+2,n-k) * binomial(3*k,k)/(2*k+1).
Showing 1-10 of 24 results. Next

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