A307678 -id:A307678 - cp's OEIS frontend

A349331 G.f. A(x) satisfies A(x) = 1 + x * A(x)^4 / (1 - x).

1, 1, 5, 31, 219, 1678, 13570, 114014, 985542, 8708099, 78298727, 714105907, 6590200215, 61427125994, 577456943614, 5468604044500, 52122539760992, 499613409224137, 4813105582181533, 46576519080852235, 452545041339982871, 4413071971740021275, 43177663974461532959

Offset: 0

Ilya Gutkovskiy, Nov 15 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..985

Cf. A002212, A002293, A307678, A317133, A346646 (partial sums), A349332, A349333, A349334, A349335.

a:= n-> coeff(series(RootOf(1+x*A^4/(1-x)-A, A), x, n+1), x, n):
seq(a(n), n=0..22);  # Alois P. Heinz, Nov 15 2021

nmax = 22; 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 - 1, k - 1] Binomial[4 k, k]/(3 k + 1), {k, 0, n}], {n, 0, 22}]

{a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0);
A[#A] = 1 + sum(k=1, m-1, (polcoeff(Ser(A)^4, k)) )); A[n+1]}
for(n=0, 30, print1(a(n), ", ")) \\ Vaclav Kotesovec, Nov 23 2024, after Paul D. Hanna

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

a(n) ~ 283^(n + 1/2) / (2^(7/2) * sqrt(Pi) * n^(3/2) * 3^(3*n + 3/2)). - Vaclav Kotesovec, Nov 15 2021

A349332 G.f. A(x) satisfies A(x) = 1 + x * A(x)^5 / (1 - x).

Original entry on oeis.org

1, 1, 6, 46, 406, 3901, 39627, 418592, 4551672, 50610692, 572807157, 6577068383, 76426719408, 897078662538, 10620634999318, 126676885170703, 1520759193166329, 18361269213121164, 222814883564042704, 2716125963857227904, 33244557641365865109

Offset: 0

Ilya Gutkovskiy, Nov 15 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..897

Cf. A002212, A307678, A349331, A349333, A349334, A349335.

Cf. A002294, A346647 (partial sums), A349361, A364748.

a:= n-> coeff(series(RootOf(1+x*A^5/(1-x)-A, A), x, n+1), x, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 15 2021

nmax = 20; 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 - 1, k - 1] Binomial[5 k, k]/(4 k + 1), {k, 0, n}], {n, 0, 20}]

{a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0);
A[#A] = 1 + sum(k=1, m-1, (polcoeff(Ser(A)^5, k)) )); A[n+1]}
for(n=0, 30, print1(a(n), ", ")) \\ Vaclav Kotesovec, Nov 23 2024, after Paul D. Hanna

a(n) = Sum_{k=0..n} binomial(n-1,k-1) * binomial(5*k,k) / (4*k+1).
a(n) ~ 3381^(n + 1/2) / (25 * sqrt(Pi) * n^(3/2) * 2^(8*n + 7/2)). - Vaclav Kotesovec, Nov 15 2021
Recurrence: 8*n*(2*n - 1)*(4*n - 1)*(4*n + 1)*a(n) = (4405*n^4 - 10346*n^3 + 9575*n^2 - 4354*n + 840)*a(n-1) - 12*(n-2)*(1255*n^3 - 3957*n^2 + 4492*n - 1820)*a(n-2) + 2*(n-3)*(n-2)*(10655*n^2 - 32733*n + 26908)*a(n-3) - 4*(n-4)*(n-3)*(n-2)*(3445*n - 6986)*a(n-4) + 3381*(n-5)*(n-4)*(n-3)*(n-2)*a(n-5). - Vaclav Kotesovec, Nov 17 2021

A349333 G.f. A(x) satisfies A(x) = 1 + x * A(x)^6 / (1 - x).

Original entry on oeis.org

1, 1, 7, 64, 678, 7836, 95838, 1219527, 15979551, 214151601, 2921712145, 40444378948, 566634504256, 8019501351103, 114484746457075, 1646614155398872, 23837794992712680, 347081039681365623, 5079306905986689309, 74670702678690897079, 1102218694940440851877

Offset: 0

Ilya Gutkovskiy, Nov 15 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..836

Cf. A002212, A307678, A349331, A349332, A349334, A349335.

Cf. A002295, A346648 (partial sums), A349362.

a:= n-> coeff(series(RootOf(1+x*A^6/(1-x)-A, A), x, n+1), x, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 15 2021

nmax = 20; 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 - 1, k - 1] Binomial[6 k, k]/(5 k + 1), {k, 0, n}], {n, 0, 20}]

{a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0);
A[#A] = 1 + sum(k=1, m-1, (polcoeff(Ser(A)^6, k)) )); A[n+1]}
for(n=0, 30, print1(a(n), ", ")) \\ Vaclav Kotesovec, Nov 23 2024, after Paul D. Hanna

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

a(n) ~ 49781^(n + 1/2) / (72 * sqrt(3*Pi) * n^(3/2) * 5^(5*n + 3/2)). - Vaclav Kotesovec, Nov 15 2021

A364983 E.g.f. satisfies A(x) = 1 + xexp(x)A(x)^3.

Original entry on oeis.org

1, 1, 8, 111, 2332, 66125, 2368086, 102616759, 5222638856, 305436798009, 20186656927210, 1488021110087171, 121044207712073196, 10771321471267219525, 1040877104088653696606, 108549742436141933697135, 12151467262433697322437136, 1453367472748861203540942065

Offset: 0

Seiichi Manyama, Aug 15 2023

nonn

Cf. A364984, A364985, A364986.
Cf. A006153, A295238, A364987.
Cf. A001764, A307678.

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

a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(3*k+1,k)/( (3*k+1)*(n-k)! ) = n! * Sum_{k=0..n} k^(n-k) * A001764(k)/(n-k)!.

a(n) ~ sqrt(3) * sqrt(1 + LambertW(4/27)) * n^(n-1) / (2^(3/2) * exp(n) * LambertW(4/27)^n). - Vaclav Kotesovec, Nov 11 2024

A349335 G.f. A(x) satisfies A(x) = 1 + x * A(x)^8 / (1 - x).

Original entry on oeis.org

1, 1, 9, 109, 1541, 23823, 390135, 6651051, 116798643, 2098313686, 38382509118, 712447023590, 13385500614902, 254065657922154, 4864482597112186, 93840443376075810, 1822169236520766546, 35586928273002974487, 698572561837366684479, 13775697096997873764647

Offset: 0

Ilya Gutkovskiy, Nov 15 2021

nonn

In general, for m > 1, Sum_{k=0..n} binomial(n-1,k-1) * binomial(m*k,k) / ((m-1)*k+1) ~ (m-1)^(m/2 - 2) * (1 + m^m/(m-1)^(m-1))^(n + 1/2) / (sqrt(2*Pi) * m^((m-1)/2) * n^(3/2)). - Vaclav Kotesovec, Nov 15 2021

Seiichi Manyama, Table of n, a(n) for n = 0..756

Cf. A002212, A307678, A349331, A349332, A349333, A349334.

Cf. A007556, A346650 (partial sums), A349364.

a:= n-> coeff(series(RootOf(1+x*A^8/(1-x)-A, A), x, n+1), x, n):
seq(a(n), n=0..19);  # Alois P. Heinz, Nov 15 2021

nmax = 19; 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 - 1, k - 1] Binomial[8 k, k]/(7 k + 1), {k, 0, n}], {n, 0, 19}]

{a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0);
A[#A] = 1 + sum(k=1, m-1, (polcoeff(Ser(A)^8, k)) )); A[n+1]}
for(n=0, 30, print1(a(n), ", ")) \\ Vaclav Kotesovec, Nov 23 2024, after Paul D. Hanna

a(n) = Sum_{k=0..n} binomial(n-1,k-1) * binomial(8*k,k) / (7*k+1).

a(n) ~ 17600759^(n + 1/2) / (2048 * sqrt(Pi) * n^(3/2) * 7^(7*n + 3/2)). - Vaclav Kotesovec, Nov 15 2021

A349334 G.f. A(x) satisfies A(x) = 1 + x * A(x)^7 / (1 - x).

Original entry on oeis.org

1, 1, 8, 85, 1051, 14197, 203064, 3022909, 46347534, 726894786, 11606936525, 188060979332, 3084087347910, 51094209834068, 853859480938095, 14376597494941454, 243649099741045190, 4153091242153905838, 71152973167920086796, 1224593757045581062444

Offset: 0

Ilya Gutkovskiy, Nov 15 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..791

Cf. A002212, A307678, A349331, A349332, A349333, A349335.

Cf. A002296, A346649 (partial sums), A349363.

a:= n-> coeff(series(RootOf(1+x*A^7/(1-x)-A, A), x, n+1), x, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 15 2021

nmax = 19; 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 - 1, k - 1] Binomial[7 k, k]/(6 k + 1), {k, 0, n}], {n, 0, 19}]

{a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0);
A[#A] = 1 + sum(k=1, m-1, (polcoeff(Ser(A)^7, k)) )); A[n+1]}
for(n=0, 30, print1(a(n), ", ")) \\ Vaclav Kotesovec, Nov 23 2024, after Paul D. Hanna

a(n) = Sum_{k=0..n} binomial(n-1,k-1) * binomial(7*k,k) / (6*k+1).

a(n) ~ 870199^(n + 1/2) / (343 * sqrt(Pi) * n^(3/2) * 2^(6*n + 2) * 3^(6*n + 3/2)). - Vaclav Kotesovec, Nov 15 2021

A371516 G.f. A(x) satisfies A(x) = (1 + x*A(x) / (1-x))^3.

Original entry on oeis.org

1, 3, 15, 82, 477, 2901, 18235, 117555, 773085, 5166478, 34987170, 239570655, 1655933060, 11538839130, 80971109712, 571702698185, 4058556404958, 28951715755830, 207424064434502, 1491898838023884, 10768487956456506, 77977009814421534, 566310026687320290

Offset: 0

Seiichi Manyama, Mar 26 2024

nonn

Cf. A045868, A371517, A371520, A371521.
Cf. A270386, A307678.
Cf. A371483.

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

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

G.f.: A(x) = B(x)^3 where B(x) is the g.f. of A307678.

A378326 a(n) = Sum_{k=0..n} binomial(n-1,k-1) * binomial(nk,k) / ((n-1)k+1).

Original entry on oeis.org

1, 1, 3, 19, 219, 3901, 95838, 3022909, 116798643, 5350403737, 283728025998, 17104314563843, 1155635807408096, 86513627563199279, 7109252862969177287, 636268582522962837475, 61610670571434193189443, 6418044336586421956746033, 715718717341021991299583730

Offset: 0

Vaclav Kotesovec, Nov 23 2024

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..338

Cf. A002212, A307678, A349331, A349332, A349333, A349334, A349335.

Cf. A377098, A378325, A378327.

Table[Sum[Binomial[n-1, k-1]*Binomial[n*k, k]/((n-1)*k+1), {k, 0, n}], {n, 0, 20}]

a(n) ~ exp(n + exp(-1) - 1/2) * n^(n - 5/2) / sqrt(2*Pi).

A379191 G.f. A(x) satisfies A(x) = (1 + xA(x))^3/(1 - xA(x)^3).

Original entry on oeis.org

1, 4, 31, 338, 4356, 61603, 923958, 14433315, 232298914, 3825260332, 64140203645, 1091364139213, 18796605318655, 327056343952311, 5740466392321499, 101516213938082457, 1807045676161156515, 32352346658163940698, 582185299986049977601, 10524395285312191583304, 191034444423571726099486

Offset: 0

Seiichi Manyama, Dec 17 2024

nonn

Cf. A379172, A379188, A379190.

Cf. A307678, A379192.

a(n) = sum(k=0, n, binomial(n+3*k+1, k)*binomial(3*n+6*k+3, n-k)/(n+3*k+1));

a(n) = Sum_{k=0..n} binomial(n+3*k+1,k) * binomial(3*n+6*k+3,n-k)/(n+3*k+1).

A366176 G.f. A(x) satisfies A(x) = 1 + x*A(x)^3/(1 - x)^2.

Original entry on oeis.org

1, 1, 5, 27, 161, 1030, 6921, 48190, 344669, 2517303, 18695908, 140771477, 1072130229, 8244820518, 63931532190, 499308229278, 3924204043333, 31012883225891, 246304580923299, 1964794017165157, 15735626383151876, 126476316316459089, 1019883740031357941

Offset: 0

Seiichi Manyama, Oct 03 2023

nonn

Partial sums give A199475.

Cf. A001764, A213282, A307678.

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

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

cp's OEIS Frontend

A349331 G.f. A(x) satisfies A(x) = 1 + x * A(x)^4 / (1 - x).

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A349332 G.f. A(x) satisfies A(x) = 1 + x * A(x)^5 / (1 - x).

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A349333 G.f. A(x) satisfies A(x) = 1 + x * A(x)^6 / (1 - x).

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A364983 E.g.f. satisfies A(x) = 1 + x*exp(x)*A(x)^3.

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A349335 G.f. A(x) satisfies A(x) = 1 + x * A(x)^8 / (1 - x).

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A349334 G.f. A(x) satisfies A(x) = 1 + x * A(x)^7 / (1 - x).

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A371516 G.f. A(x) satisfies A(x) = (1 + x*A(x) / (1-x))^3.

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A378326 a(n) = Sum_{k=0..n} binomial(n-1,k-1) * binomial(n*k,k) / ((n-1)*k+1).

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A364983 E.g.f. satisfies A(x) = 1 + xexp(x)A(x)^3.

A378326 a(n) = Sum_{k=0..n} binomial(n-1,k-1) * binomial(nk,k) / ((n-1)k+1).

A379191 G.f. A(x) satisfies A(x) = (1 + xA(x))^3/(1 - xA(x)^3).