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 10 results.

A346626 G.f. A(x) satisfies: A(x) = (1 + x * A(x)^3) / (1 - x).

Original entry on oeis.org

1, 2, 8, 44, 280, 1936, 14128, 107088, 834912, 6652608, 53934080, 443467136, 3689334272, 30997608960, 262651640064, 2241857334528, 19257951946240, 166362924583936, 1444351689281536, 12595885932259328, 110287974501355520, 969178569410404352, 8544982917273509888, 75565732555028701184
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 25 2021

Keywords

Comments

Partial sums of A213282.

Links

Crossrefs

Cf. A001764, A006318, A199475, A213282, A349310, A349311, A349312, A349313, A349314.

Programs

  • Mathematica
    nmax = 23; A[] = 0; Do[A[x] = (1 + x A[x]^3)/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
nmax = 23; CoefficientList[Series[Sum[(Binomial[3 k, k]/(2 k + 1)) x^k/(1 - x)^(3 k + 1), {k, 0, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = a[n - 1] + 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, 23}]

Formula

G.f.: Sum_{k>=0} ( binomial(3*k,k) / (2*k + 1) ) * x^k / (1 - x)^(3*k+1).
a(0) = 1; a(n) = a(n-1) + Sum_{i=0..n-1} Sum_{j=0..n-i-1} a(i) * a(j) * a(n-i-j-1).
a(n) ~ 2^(n - 1/2) / (sqrt(3*Pi*(2 - (2 - sqrt(2))^(1/3)/2^(2/3) - 1/(2*(2 - sqrt(2)))^(1/3))) * n^(3/2) * (2 - 3/(sqrt(2) - 1)^(1/3) + 3*(sqrt(2) - 1)^(1/3))^n). - Vaclav Kotesovec, Nov 04 2021
a(n) = (1/n) * Sum_{k=0..floor((n-1)/2)} 2^(n-k) * binomial(n,k) * binomial(2*n-k,n-1-2*k) for n > 0. - Seiichi Manyama, Apr 01 2024

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

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

A366401 G.f. A(x) satisfies A(x) = (1 + x * A(x)^(7/2)) / (1 - x).

Original entry on oeis.org

1, 2, 9, 58, 436, 3572, 30935, 278532, 2581043, 24453404, 235790159, 2306367444, 22829030276, 228240387070, 2301498462245, 23379656027868, 239038022347243, 2457891085704180, 25400777844198274, 263685720722690420, 2748421883496133866
Offset: 0

Views

Author

Seiichi Manyama, Oct 09 2023

Keywords

Crossrefs

Cf. A349313, A366400, A366402, A366403, A366404, A366405, A366406, A366407.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(n+5*k/2, n-k)*binomial(7*k/2, k)/(5*k/2+1));

Formula

a(n) = Sum_{k=0..n} binomial(n+5*k/2,n-k) * binomial(7*k/2,k) / (5*k/2+1).

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).

A366366 G.f. satisfies A(x) = (1 + x/A(x)^4)/(1 - x).

Original entry on oeis.org

1, 2, -6, 58, -574, 6402, -75878, 939290, -12000318, 157050178, -2094657926, 28368411194, -389079656446, 5393118559938, -75431624084838, 1063251390845338, -15088643098754942, 215396586102923138, -3091050571516120582, 44566089825496186170
Offset: 0

Views

Author

Seiichi Manyama, Oct 08 2023

Keywords

Crossrefs

Cf. A346626, A349310, A349311, A349312, A349313, A349314, A366363, A366364, A366365.
Cf. A366359.

Programs

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

Formula

a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(5*k-1,k) * binomial(4*k-1,n-k)/(5*k-1).
Showing 1-10 of 10 results.

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