A349312 -id:A349312 - cp's OEIS frontend

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

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

Ilya Gutkovskiy, Jul 25 2021

nonn

Partial sums of A213282.

Seiichi Manyama, Table of n, a(n) for n = 0..500
Paul Barry, Notes on Riordan arrays and lattice paths, arXiv:2504.09719 [math.CO], 2025. See p. 26.

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

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}]

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

Ilya Gutkovskiy, Nov 14 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..500

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

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}]

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

Ilya Gutkovskiy, Nov 14 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..500

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

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}]

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

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

Ilya Gutkovskiy, Nov 14 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..500

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

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}]

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

Seiichi Manyama, Oct 08 2023

sign

Paolo Xausa, Table of n, a(n) for n = 0..1000

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

Cf. A366356.

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 *)

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

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

Seiichi Manyama, Oct 08 2023

sign

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

Cf. A366357.

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

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

Ilya Gutkovskiy, Nov 14 2021

nonn

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

Seiichi Manyama, Table of n, a(n) for n = 0..500

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

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}]

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

A364409 G.f. satisfies A(x) = 1 + x*(1 + 1/A(x)^5).

Original entry on oeis.org

1, 2, -10, 110, -1430, 20570, -315282, 5047350, -83406510, 1411954610, -24360750810, 426796726334, -7572551327430, 135790011411850, -2457028916693090, 44804882306441990, -822573909558939998, 15191515999168557410, -282038057756813698730

Offset: 0

Seiichi Manyama, Jul 23 2023

sign

Seiichi Manyama, Table of n, a(n) for n = 0..769

Cf. A112478, A364393, A364407, A364408, A366266, A366267, A366268.

Cf. A349312.

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

G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A349312.

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

A364523 G.f. satisfies A(x) = 1 + xA(x) + x^6A(x)^6.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 8, 29, 85, 211, 463, 931, 1795, 3550, 7736, 18929, 49505, 130000, 330430, 804271, 1885675, 4327555, 9929515, 23224435, 55907251, 138016906, 345107296, 862546231, 2136402451, 5231163232, 12697101118, 30723857209, 74569942745

Offset: 0

Seiichi Manyama, Jul 27 2023

nonn

Number of ordered trees with n edges and having nonleaf nodes of outdegrees 1 or 6. - Emanuele Munarini, Jul 11 2024

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A001006, A071879, A127902, A364522.

Cf. A349312, A364472.

a(n) = sum(k=0, n\6, binomial(n, 6*k)*binomial(6*k, k)/(5*k+1));

a(n) = Sum_{k=0..floor(n/6)} binomial(n,6*k) * binomial(6*k,k) / (5*k+1).

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

Seiichi Manyama, Oct 08 2023

sign

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

Cf. A366358.

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

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

cp's OEIS Frontend

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

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

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

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

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

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

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

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A364409 G.f. satisfies A(x) = 1 + x*(1 + 1/A(x)^5).

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A364523 G.f. satisfies A(x) = 1 + x*A(x) + x^6*A(x)^6.

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A364523 G.f. satisfies A(x) = 1 + xA(x) + x^6A(x)^6.