A349332 -id:A349332 - 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

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

A349361 G.f. A(x) satisfies: A(x) = 1 + x * A(x)^5 / (1 + x).

Original entry on oeis.org

1, 1, 4, 26, 194, 1581, 13625, 122120, 1126780, 10631460, 102104845, 994855179, 9809872626, 97710157154, 981636609906, 9935473707279, 101214412755647, 1036991125300748, 10678412226507032, 110459290208905008, 1147261657267290037

Offset: 0

Ilya Gutkovskiy, Nov 15 2021

nonn

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

Cf. A001006, A002294, A127897, A317133, A345368 (binomial transform), A346665, A349332, A349362, A349363, A349364.

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

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n-1,k-1) * binomial(5*k,k) / (4*k+1).
a(n) = (-1)^(n+1)*F([6/5, 7/5, 8/5, 9/5, 1-n], [3/2, 7/4, 2, 9/4], 5^5/2^8), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 15 2021
From Vaclav Kotesovec, Nov 17 2021: (Start)
a(n) ~ 2869^(n + 1/2) / (25 * sqrt(Pi) * n^(3/2) * 2^(8*n + 7/2)).
Recurrence: 8*n*(2*n - 1)*(4*n - 1)*(4*n + 1)*a(n) = 3*(615*n^4 - 718*n^3 - 275*n^2 + 618*n - 200)*a(n-1) + 4*(n-2)*(2485*n^3 - 6879*n^2 + 6524*n - 2040)*a(n-2) + 2*(n-3)*(n-2)*(8095*n^2 - 23517*n + 18092)*a(n-3) + 12*(n-4)*(n-3)*(n-2)*(935*n - 1838)*a(n-4) + 2869*(n-5)*(n-4)*(n-3)*(n-2)*a(n-5). (End)

A365184 G.f. satisfies A(x) = 1 + xA(x)^5(1 + x).

Original entry on oeis.org

1, 1, 6, 45, 395, 3775, 38146, 400826, 4335455, 47951065, 539823620, 6165377836, 71261299056, 831990025420, 9797505040130, 116235417614900, 1387958781395535, 16668362761081560, 201190667288072005, 2439418470063468505, 29698136499328762445

Offset: 0

Seiichi Manyama, Aug 25 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..898

Cf. A002295, A365185, A365186, A365187, A365188, A365189.
Cf. A364475, A365178.
Cf. A002294, A349332.

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

a(n) = Sum_{k=0..n} binomial(5*k+1,k) * binomial(k,n-k)/(5*k+1) = Sum_{k=0..n} binomial(k,n-k) * A002294(k).

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

A365192 G.f. satisfies A(x) = 1 + xA(x)^5 / (1 - xA(x)^2).

Original entry on oeis.org

1, 1, 6, 48, 443, 4445, 47107, 518835, 5880223, 68130860, 803369481, 9609294542, 116310009888, 1421951861817, 17533301767624, 217796367181117, 2722942699583650, 34236790400004432, 432649744252128084, 5492060945760586212, 69998993052214823013

Offset: 0

Seiichi Manyama, Aug 25 2023

nonn

Cf. A002295, A243667, A349332, A364748, A365193, A365194.

Cf. A365186.

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

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

A371486 G.f. A(x) satisfies A(x) = 1 / (1 - x*A(x) / (1-x))^4.

Original entry on oeis.org

1, 4, 30, 260, 2465, 24796, 260008, 2811216, 31117240, 350890260, 4016744586, 46556054072, 545273713228, 6443442857024, 76727957438650, 919796418086076, 11091249210406816, 134439965189940176, 1637160457090585016, 20019920157735604796, 245733987135102838131

Offset: 0

Seiichi Manyama, Mar 25 2024

nonn

Cf. A002212, A270386, A371483.

Cf. A118971, A349332.

a(n) = sum(k=0, n, binomial(n-1, n-k)*binomial(5*k+3, k)/(k+1));

a(n) = Sum_{k=0..n} binomial(n-1,n-k) * binomial(5*k+3,k)/(k+1).
G.f.: A(x) = B(x/(1-x)), where B(x) = (1/x) * Series_Reversion( x*(1-x)^4 ).
G.f.: A(x) = B(x)^4 where B(x) is the g.f. of A349332.

A365193 G.f. satisfies A(x) = 1 + xA(x)^5 / (1 - xA(x)^3).

Original entry on oeis.org

1, 1, 6, 49, 463, 4760, 51702, 583712, 6781774, 80555066, 973813974, 11941861079, 148191437719, 1857464450449, 23481830726334, 299056887494427, 3833349330581255, 49416395972195630, 640256115370243620, 8332835556325119938, 108890550249605779116

Offset: 0

Seiichi Manyama, Aug 25 2023

nonn

Cf. A002295, A243667, A349332, A364748, A365192, A365194.

Cf. A365187.

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

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

A371520 G.f. A(x) satisfies A(x) = (1 + x*A(x) / (1-x))^5.

Original entry on oeis.org

1, 5, 40, 360, 3495, 35726, 378965, 4133080, 46059020, 522196465, 6004261226, 69849651025, 820651943130, 9723556336780, 116056250171385, 1394082307995626, 16840510019954835, 204453614350921540, 2493311080293185200, 30528431677508637205, 375155454309681439001

Offset: 0

Seiichi Manyama, Mar 26 2024

nonn

Cf. A045868, A371516, A371517, A371521.

Cf. A349332, A371486.

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

a(n) = 5 * Sum_{k=0..n} binomial(n-1,n-k) * binomial(5*k+4,k)/(4*k+5) = Sum_{k=0..n} binomial(n-1,n-k) * binomial(5*k+5,k)/(k+1).

G.f.: A(x) = B(x)^5 where B(x) is the g.f. of A349332.

cp's OEIS Frontend

A349331 G.f. A(x) satisfies A(x) = 1 + x * A(x)^4 / (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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A349361 G.f. A(x) satisfies: A(x) = 1 + x * A(x)^5 / (1 + x).

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A365184 G.f. satisfies A(x) = 1 + x*A(x)^5*(1 + x).

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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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A365192 G.f. satisfies A(x) = 1 + x*A(x)^5 / (1 - x*A(x)^2).

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A371486 G.f. A(x) satisfies A(x) = 1 / (1 - x*A(x) / (1-x))^4.

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A365184 G.f. satisfies A(x) = 1 + xA(x)^5(1 + x).

A365192 G.f. satisfies A(x) = 1 + xA(x)^5 / (1 - xA(x)^2).

A365193 G.f. satisfies A(x) = 1 + xA(x)^5 / (1 - xA(x)^3).