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 27 results. Next

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

A349289 G.f. A(x) satisfies A(x) = 1 / ((1 - x) * (1 - x * A(x)^3)).

Original entry on oeis.org

1, 2, 9, 61, 493, 4371, 41065, 401563, 4044097, 41658044, 436862457, 4648331765, 50057856881, 544557984498, 5975422922413, 66059269445451, 735064865871889, 8226310738656892, 92531697191189777, 1045551973586825023, 11862334695799444993
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 13 2021

Keywords

Links

Crossrefs

Cf. A002293, A007317, A199475, A346646, A349290, A349291, A349292, A349293.

Programs

  • Maple
    A349289 := proc(n)
    add( binomial(n+2*k,3*k)*binomial(4*k,k)/(3*k+1),k=0..n)  ;
end proc:
seq(A349289(n),n=0..50) ; # R. J. Mathar, Feb 10 2024
  • Mathematica
    nmax = 20; A[] = 0; Do[A[x] = 1/((1 - x) (1 - x A[x]^3)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n + 2 k, 3 k] Binomial[4 k, k]/(3 k + 1), {k, 0, n}], {n, 0, 20}]

Formula

a(n) = Sum_{k=0..n} binomial(n+2*k,3*k) * binomial(4*k,k) / (3*k+1).
a(n) = F([1/4, 1/2, 3/4, (1+n)/2, (2+n)/2, -n], [1/3, 2/3, 2/3, 1, 4/3], -2^10/3^6) where F is the generalized hypergeometric function. - Stefano Spezia, Nov 13 2021
a(n) ~ sqrt(1 + 2*r) / (4 * 2^(1/6) * sqrt(3*Pi*(1-r)) * n^(3/2) * r^(n + 1/3)), where r = 0.0816785448577670972635343365300887975661075663022821172271... is the root of the equation 4^4 * r = 3^3 * (1-r)^3. - Vaclav Kotesovec, Nov 14 2021
D-finite with recurrence 81*n*(3*n-1)*(3*n+1)*a(n) +3*(243*n^3-7101*n^2+9986*n-3560)*a(n-1) +(-115027*n^3+514908*n^2-699869*n+269580)*a(n-2) +(-85543*n^3+1604715*n^2-6291692*n+6995280)*a(n-3) +(580211*n^3-6643158*n^2+23063299*n-23830944)*a(n-4) +(-33473*n^3-2231073*n^2+26352470*n-70945392)*a(n-5) +(-872129*n^3+17812344*n^2-119542699*n+264170868)*a(n-6) +(667171*n^3-14196243*n^2+100393472*n-236010000)*a(n-7) -6*(3*n-23)*(9948*n^2-147805*n+548868)*a(n-8) +4044*(3*n-26)*(n-8)*(3*n-22)*a(n-9)=0. - R. J. Mathar, Feb 10 2024
a(n) = 1 + Sum_{i, j, k, l>=0 and i+j+k+l=n-1} a(i) * a(j) * a(k) * a(l). - Seiichi Manyama, Jul 10 2025

A349290 G.f. A(x) satisfies A(x) = 1 / ((1 - x) * (1 - x * A(x)^4)).

Original entry on oeis.org

1, 2, 11, 96, 1001, 11456, 139013, 1756596, 22867421, 304560171, 4130200726, 56836946342, 791689962811, 11140615233281, 158140107648676, 2261708608884896, 32559326010349817, 471428798399646336, 6860801662510005266, 100302910051255600486
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 13 2021

Keywords

Links

Crossrefs

Cf. A002294, A007317, A199475, A346647, A349289, A349291, A349292, A349293.

Programs

  • Mathematica
    nmax = 19; A[] = 0; Do[A[x] = 1/((1 - x) (1 - x A[x]^4)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n + 3 k, 4 k] Binomial[5 k, k]/(4 k + 1), {k, 0, n}], {n, 0, 19}]
  • PARI
    a(n) = sum(k=0, n, binomial(n+3*k,4*k) * binomial(5*k,k) / (4*k+1)); \\ Michel Marcus, Nov 14 2021

Formula

a(n) = Sum_{k=0..n} binomial(n+3*k,4*k) * binomial(5*k,k) / (4*k+1).
a(n) = F([1/5, 2/5, 3/5, 4/5, (1+n)/3, (2+n)/3, (3+n)/3, -n], [1/4, 1/2, 1/2, 3/4, 3/4, 1, 5/4], -3^3*5^5/2^16), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 13 2021
a(n) ~ sqrt(1 + 3*r) / (2 * 5^(3/4) * sqrt(2*Pi*(1-r)) * n^(3/2) * r^(n + 1/4)), where r = 0.0631152861998150860738633360987635931... is the root of the equation 5^5 * r = 4^4 * (1-r)^4. - Vaclav Kotesovec, Nov 14 2021
a(n) = 1 + Sum_{i, j, k, l, m>=0 and i+j+k+l+m=n-1} a(i) * a(j) * a(k) * a(l) * a(m). - Seiichi Manyama, Jul 10 2025

A349291 G.f. A(x) satisfies A(x) = 1 / ((1 - x) * (1 - x * A(x)^5)).

Original entry on oeis.org

1, 2, 13, 139, 1775, 24886, 370099, 5733304, 91518691, 1494815215, 24862931821, 419674102147, 7170713484877, 123783319369420, 2155542171446485, 37820343323942566, 667957770644685811, 11865421405897931581, 211856917750711562695, 3800040255017879663415
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 13 2021

Keywords

Links

Crossrefs

Cf. A002295, A007317, A199475, A346648, A349289, A349290, A349292, A349293.

Programs

  • Mathematica
    nmax = 19; A[] = 0; Do[A[x] = 1/((1 - x) (1 - x A[x]^5)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n + 4 k, 5 k] Binomial[6 k, k]/(5 k + 1), {k, 0, n}], {n, 0, 19}]

Formula

a(n) = Sum_{k=0..n} binomial(n+4*k,5*k) * binomial(6*k,k) / (5*k+1).
a(n) ~ sqrt(1 + 4*r) / (2^(6/5) * 3^(7/10) * sqrt(5*Pi*(1-r)) * n^(3/2) * r^(n + 1/5)), where r = 0.051436794119208432185504972091697516647... is the real root of the equation 6^6 * r = 5^5 * (1-r)^5. - Vaclav Kotesovec, Nov 14 2021
a(n) = 1 + Sum_{x_1, x_2, ..., x_6>=0 and x_1+x_2+...+x_6=n-1} Product_{k=1..6} a(x_k). - Seiichi Manyama, Jul 10 2025

A349292 G.f. A(x) satisfies A(x) = 1 / ((1 - x) * (1 - x * A(x)^6)).

Original entry on oeis.org

1, 2, 15, 190, 2871, 47643, 838888, 15389452, 290951545, 5629024955, 110908062511, 2217739684483, 44891645810124, 918086053852234, 18941156419798530, 393742848618632760, 8239112912485293357, 173406208518520952066, 3668419587671991125142
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 13 2021

Keywords

Links

Crossrefs

Cf. A002296, A007317, A199475, A346649, A349289, A349290, A349291, A349293.

Programs

  • Mathematica
    nmax = 18; A[] = 0; Do[A[x] = 1/((1 - x) (1 - x A[x]^6)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n + 5 k, 6 k] Binomial[7 k, k]/(6 k + 1), {k, 0, n}], {n, 0, 18}]

Formula

a(n) = Sum_{k=0..n} binomial(n+5*k,6*k) * binomial(7*k,k) / (6*k+1).
a(n) ~ sqrt(1 + 5*r) / (2 * 7^(2/3) * sqrt(3*Pi*(1-r)) * n^(3/2) * r^(n + 1/6)), where r = 0.043408935906208378827553096713877784793679356... is the root of the equation 7^7 * r = 6^6 * (1-r)^6. - Vaclav Kotesovec, Nov 14 2021
a(n) = 1 + Sum_{x_1, x_2, ..., x_7>=0 and x_1+x_2+...+x_7=n-1} Product_{k=1..7} a(x_k). - Seiichi Manyama, Jul 11 2025

A349293 G.f. A(x) satisfies A(x) = 1 / ((1 - x) * (1 - x * A(x)^7)).

Original entry on oeis.org

1, 2, 17, 249, 4345, 83285, 1694273, 35915349, 784691569, 17545398747, 399545961817, 9234298584921, 216053290499201, 5107287712887563, 121795876378121121, 2926604574330886897, 70788399943851406825, 1722188546498276868124, 42114624858397590035177
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 13 2021

Keywords

Comments

In general, for k>=1, Sum_{j=0..n} binomial(n + (k-1)*j,k*j) * binomial((k+1)*j,j) / (k*j+1) ~ sqrt(1 + (k-1)*r) / ((k+1)^(1/k) * sqrt(2*k*(k+1)*Pi*(1-r)) * n^(3/2) * r^(n + 1/k)), where r is the smallest real root of the equation (k+1)^(k+1) * r = k^k * (1-r)^k. - Vaclav Kotesovec, Nov 14 2021

Links

Crossrefs

Cf. A007317, A007556, A199475, A346650, A349289, A349290, A349291, A349292.

Programs

  • Mathematica
    nmax = 18; A[] = 0; Do[A[x] = 1/((1 - x) (1 - x A[x]^7)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[Binomial[n + 6 k, 7 k] Binomial[8 k, k]/(7 k + 1), {k, 0, n}], {n, 0, 18}]
  • PARI
    a(n) = sum(k=0, n, binomial(n+6*k,7*k) * binomial(8*k,k) / (7*k+1)); \\ Michel Marcus, Nov 14 2021

Formula

a(n) = Sum_{k=0..n} binomial(n+6*k,7*k) * binomial(8*k,k) / (7*k+1).
a(n) ~ sqrt(1 + 6*r) / (2^(17/7) * sqrt(7*Pi*(1-r)) * n^(3/2) * r^(n + 1/7)), where r = 0.0375502499742240443056934699070050852345109331376051496159609551... is the real root of the equation 8^8 * r = 7^7 * (1-r)^7. - Vaclav Kotesovec, Nov 14 2021
a(n) = 1 + Sum_{x_1, x_2, ..., x_8>=0 and x_1+x_2+...+x_8=n-1} Product_{k=1..8} a(x_k). - Seiichi Manyama, Jul 11 2025

A346627 G.f. A(x) satisfies: A(x) = 1 / (1 + x) + x * A(x)^3.

Original entry on oeis.org

1, 0, 1, 2, 7, 23, 82, 300, 1129, 4334, 16914, 66899, 267586, 1080516, 4398850, 18035084, 74402361, 308624282, 1286428765, 5385578256, 22635057148, 95471113565, 403983783772, 1714494024947, 7295949019114, 31124885587680, 133085594104222, 570266646942488
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 25 2021

Keywords

Comments

Inverse binomial transform of A200753.

Links

Crossrefs

Cf. A001764, A005043, A199475, A200753, A346628, A349299, A349300, A349301, A349302, A349303.

Programs

  • Mathematica
    nmax = 27; A[] = 0; Do[A[x] = 1/(1 + x) + x A[x]^3 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = (-1)^n + 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, 27}]
Table[Sum[(-1)^(n - k) Binomial[n + k, n - k] Binomial[3 k, k]/(2 k + 1), {k, 0, n}], {n, 0, 27}]

Formula

G.f.: Sum_{k>=0} ( binomial(3*k,k) / (2*k + 1) ) * x^k / (1 + x)^(2*k+1).
a(n) = (-1)^n + Sum_{i=0..n-1} Sum_{j=0..n-i-1} a(i) * a(j) * a(n-i-j-1).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n+k,n-k) * binomial(3*k,k) / (2*k + 1).
a(n) ~ sqrt(198 + 38*sqrt(33)) * (19 + 3*sqrt(33))^n / (9 * sqrt(Pi) * n^(3/2) * 2^(3*n + 3)). - Vaclav Kotesovec, Jul 30 2021

A364623 G.f. satisfies A(x) = 1/(1-x)^3 + x*A(x)^3.

Original entry on oeis.org

1, 4, 18, 112, 847, 7086, 62974, 583002, 5560323, 54249583, 538873135, 5431177821, 55402340842, 570899082760, 5933922697380, 62138800690564, 654949976467593, 6942859160218698, 73972792893687427, 791722414873487767, 8508265804914763731
Offset: 0

Views

Author

Seiichi Manyama, Jul 30 2023

Keywords

Links

Crossrefs

Partial sums of A364629.
Cf. A162481, A364624.
Cf. A001764, A199475, A364620.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(n+5*k+2, 6*k+2)*binomial(3*k, k)/(2*k+1));

Formula

a(n) = Sum_{k=0..n} binomial(n+5*k+2,6*k+2) * binomial(3*k,k) / (2*k+1).

A349253 G.f. A(x) satisfies A(x) = 1 / ((1 - x) * (1 - 2 * x * A(x)^2)).

Original entry on oeis.org

1, 3, 19, 169, 1753, 19795, 236035, 2923857, 37256881, 485202307, 6429346899, 86405569657, 1174917167881, 16134949855251, 223460304878467, 3117521211476641, 43771643214792033, 618045740600046211, 8770377489446217235, 125013010654218317385, 1789104455068153153849
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 12 2021

Keywords

Links

Crossrefs

Cf. A001764, A103210, A153231, A162326, A199475, A349254, A349255, A349256.

Programs

  • Mathematica
    nmax = 20; A[] = 0; Do[A[x] = 1/((1 - x) (1 - 2 x A[x]^2)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = 1 + 2 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, 20}]
Table[Sum[Binomial[n + k, n - k] 2^k Binomial[3 k, k]/(2 k + 1), {k, 0, n}], {n, 0, 20}]
a[n_] := HypergeometricPFQ[{1/3,2/3,-n,n + 1}, {1/2,1,3/2}, -(3/2)^3];
Table[a[n], {n, 0, 20}] (* Peter Luschny, Nov 12 2021 *)

Formula

a(n) = 1 + 2 * Sum_{i=0..n-1} Sum_{j=0..n-i-1} a(i) * a(j) * a(n-i-j-1).
a(n) = Sum_{k=0..n} binomial(n+k,n-k) * 2^k * binomial(3*k,k) / (2*k+1).
a(n) = hypergeom([1/3, 2/3, -n, n + 1], [1/2, 1, 3/2], -(3/2)^3). - Peter Luschny, Nov 12 2021
a(n) ~ sqrt(315 + 31*sqrt(105)) * (31 + 3*sqrt(105))^n / (9 * sqrt(Pi) * 2^(2*n + 5/2) * n^(3/2)). - Vaclav Kotesovec, Nov 13 2021

A349254 G.f. A(x) satisfies A(x) = 1 / ((1 - x) * (1 - 3 * x * A(x)^2)).

Original entry on oeis.org

1, 4, 37, 478, 7159, 116497, 2000386, 35671756, 654218641, 12261271942, 233798163646, 4521194100541, 88458184054882, 1747850650032532, 34828329987024058, 699083528482636228, 14121906499195594537, 286877562430915732546, 5856866441794110926809
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 12 2021

Keywords

Links

Crossrefs

Cf. A001764, A103211, A199475, A217363, A337167, A349253, A349255, A349256.

Programs

  • Mathematica
    nmax = 18; A[] = 0; Do[A[x] = 1/((1 - x) (1 - 3 x A[x]^2)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = 1 + 3 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, 18}]
Table[Sum[Binomial[n + k, n - k] 3^k Binomial[3 k, k]/(2 k + 1), {k, 0, n}], {n, 0, 18}]
a[n_] := HypergeometricPFQ[{1/3, 2/3, -n, n + 1}, {1/2, 1, 3/2}, -81/16];
Table[a[n], {n, 0, 18}] (* Peter Luschny, Nov 12 2021 *)

Formula

a(n) = 1 + 3 * Sum_{i=0..n-1} Sum_{j=0..n-i-1} a(i) * a(j) * a(n-i-j-1).
a(n) = Sum_{k=0..n} binomial(n+k,n-k) * 3^k * binomial(3*k,k) / (2*k+1).
a(n) = hypergeom([1/3, 2/3, -n, n + 1], [1/2, 1, 3/2], -(3/2)^4). - Peter Luschny, Nov 12 2021
a(n) ~ sqrt(873 + 89*sqrt(97)) * (89 + 9*sqrt(97))^n / (3^(5/2) * sqrt(Pi) * n^(3/2) * 2^(3*n + 5/2)). - Vaclav Kotesovec, Nov 13 2021
Showing 1-10 of 27 results. Next

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