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.

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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

Views

Author

Ilya Gutkovskiy, Nov 15 2021

Keywords

Links

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    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}]

Formula

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)

A118968 a(4n+k) = (k+1)*binomial(5n+k,n)/(4n+k+1), k=0..3.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 4, 5, 11, 18, 26, 35, 80, 136, 204, 285, 665, 1155, 1771, 2530, 5980, 10530, 16380, 23751, 56637, 100688, 158224, 231880, 556512, 996336, 1577532, 2330445, 5620485, 10116873, 16112057, 23950355, 57985070, 104819165, 167710664, 250543370, 608462470
Offset: 0

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Author

Paul Barry, May 07 2006

Keywords

Comments

Row sums of Riordan array (1,x(1-x^4))^(-1).

Links

Crossrefs

Cf. A124753, A002294, A118969, A118970, A118971.

Programs

  • Mathematica
    Table[k=Mod[n,4];(k+1)Binomial[(5n-k)/4,(n-k)/4]/(n+1),{n,0,40}] (* Robert A. Russell, Mar 14 2024 *)
  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+x*A^2*subst(A,x,-x)*subst(A,x,I*x)*subst(A,x,-I*x));polcoeff(A,n)} \\ Paul D. Hanna, Jun 04 2012
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1+x*A*exp(sum(m=1,n\4,4*polcoeff(log(A+x*O(x^n)),4*m)*x^(4*m))+x*O(x^n)));polcoeff(A,n)} \\ Paul D. Hanna, Jun 04 2012
  • PARI
    apr(n, p, r) = r*binomial(n*p+r, n)/(n*p+r);
a(n) = apr(n\4, 5, n%4+1); \\ Seiichi Manyama, Jul 20 2025

Formula

a(4n) = A002294(n), a(4n+1) = A118969(n), a(4n+2) = A118970(n), a(4n+3) = A118971(n).
G.f. satisfies: A(x) = 1 + x*A(x)^2*A(-x)*A(I*x)*A(-I*x). - Paul D. Hanna, Jun 04 2012
G.f. satisfies: A(x) = 1 + x*A(x)*G(x^4) where G(x) = 1 + x*G(x)^5 is the g.f. of A002294. - Paul D. Hanna, Jun 04 2012
From Robert A. Russell, Mar 14 2024: (Start)
G.f.: G(z^4) + z*G(z^4)^2 + z^2*G(z^4)^3 + z^3*G(z^4)^4, where G(z) = 1 + z*G(z)^5 is the g.f. for A002294.
G.f.: E(1)(t*E(5)(t^4)) (fifth entry in Table 3), where E(d)(t) is defined in formula 3 of Hering link. (End)
a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/4)} a(4*k) * a(n-1-4*k). - Seiichi Manyama, Jul 07 2025

A346665 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(5*k,k) / (4*k + 1).

Original entry on oeis.org

1, 0, 4, 22, 172, 1409, 12216, 109904, 1016876, 9614584, 92490261, 902364918, 8907507708, 88802649446, 892833960460, 9042639746819, 92171773008828, 944819352291920, 9733592874215112, 100725697334689896, 1046535959932600141, 10913073121311627481, 114175868855824821752
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 27 2021

Keywords

Comments

Inverse binomial transform of A002294.

Links

Crossrefs

Cf. A002294, A005043, A346628, A346647, A346664, A346666, A346667, A346668.

Programs

  • Maple
    A346665 := proc(n)
    add((-1)^(n-k)*binomial(n,k)*binomial(5*k,k)/(4*k+1),k=0..n) ;
end proc:
seq(A346665(n),n=0..80); # R. J. Mathar, Aug 17 2023
  • Mathematica
    Table[Sum[(-1)^(n - k) Binomial[n, k] Binomial[5 k, k]/(4 k + 1), {k, 0, n}], {n, 0, 22}]
nmax = 22; A[] = 0; Do[A[x] = 1/(1 + x) + x (1 + x)^3 A[x]^5 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
nmax = 22; CoefficientList[Series[Sum[(Binomial[5 k, k]/(4 k + 1)) x^k/(1 + x)^(k + 1), {k, 0, nmax}], {x, 0, nmax}], x]
Table[(-1)^n HypergeometricPFQ[{1/5, 2/5, 3/5, 4/5, -n}, {1/2, 3/4, 1, 5/4}, 3125/256], {n, 0, 22}]
  • PARI
    a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n,k)*binomial(5*k,k)/(4*k + 1)); \\ Michel Marcus, Jul 28 2021

Formula

G.f. A(x) satisfies: A(x) = 1 / (1 + x) + x * (1 + x)^3 * A(x)^5.
G.f.: Sum_{k>=0} ( binomial(5*k,k) / (4*k + 1) ) * x^k / (1 + x)^(k+1).
a(n) ~ 2869^(n + 3/2) / (78125 * sqrt(Pi) * n^(3/2) * 2^(8*n + 7/2)). - Vaclav Kotesovec, Jul 30 2021
D-finite with recurrence 8*n*(4*n+1)*(2*n-1)*(4*n-1)*a(n) -(n-1) *(1845*n^3 -1333*n^2 -238*n +240)*a(n-1) -4*(n-1) *(2485*n^3 -7263*n^2 +7388*n -2580) *a(n-2) -2*(n-1) *(n-2) *(8095*n^2 -24029*n +18924) *a(n-3) -4*(n-1) *(n-2) *(n-3) *(2805*n -5578) *a(n-4) -2869*(n-1) *(n-2) *(n-3) *(n-4) *a(n-5)=0. - R. J. Mathar, Aug 17 2023

A364747 G.f. A(x) satisfies A(x) = 1 + x*A(x)^4 / (1 - x*A(x)).

Original entry on oeis.org

1, 1, 5, 32, 234, 1854, 15490, 134380, 1198944, 10931761, 101412677, 954155059, 9083120975, 87326765375, 846709605539, 8269910074087, 81291388929027, 803592049667495, 7983612883739843, 79671910265120574, 798283229227457304, 8027625597750959053
Offset: 0

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Author

Seiichi Manyama, Aug 05 2023

Keywords

Links

Crossrefs

Cf. A000108, A001003, A108447, A364748, A378691, A378692.
Cf. A002294, A243659, A364765, A364792.

Programs

  • PARI
    a(n) = if(n==0, 1, sum(k=0, n-1, binomial(n, k)*binomial(4*n-3*k, n-1-k))/n);
  • PARI
    a(n, r=1, s=1, t=4, u=1) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(n+(s-1)*k-1, n-k)/(t*k+u*(n-k)+r)); \\ Seiichi Manyama, Dec 05 2024

Formula

a(n) = (1/n) * Sum_{k=0..n-1} binomial(n,k) * binomial(4*n-3*k,n-1-k) for n > 0.
From Seiichi Manyama, Dec 05 2024: (Start)
G.f. A(x) satisfies A(x) = 1/(1 - x*A(x)^3/(1 - x*A(x))).
If g.f. satisfies A(x) = ( 1 + x*A(x)^(t/r) / (1 - x*A(x)^(u/r))^s )^r, then a(n) = r * Sum_{k=0..n} binomial(t*k+u*(n-k)+r,k) * binomial(n+(s-1)*k-1,n-k)/(t*k+u*(n-k)+r). (End)

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

Views

Author

Seiichi Manyama, Aug 25 2023

Keywords

Links

Crossrefs

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

Programs

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

Formula

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).

A079589 a(n) = C(5*n+1,n).

Original entry on oeis.org

1, 6, 55, 560, 5985, 65780, 736281, 8347680, 95548245, 1101716330, 12777711870, 148902215280, 1742058970275, 20448884000160, 240719591939480, 2840671544105280, 33594090947249085, 398039194165652550, 4724081931321677925, 56151322242892212960, 668324943343021950370
Offset: 0

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Author

Benoit Cloitre, Jan 26 2003

Keywords

Comments

a(n) is the number of paths from (0,0) to (5n,n) taking north and east steps while avoiding exactly 2 consecutive north steps. - Shanzhen Gao, Apr 15 2010

Links

Crossrefs

Cf. A052203.
Cf. A001449, A079678, A371753, A385632, A386812.
Cf. A005810, A183160.

Programs

  • Magma
    [Binomial(5*n+1, n): n in [0..20]]; // Vincenzo Librandi, Aug 07 2014
  • Maple
    seq(binomial(5*n+1,n),n=0..100); # Robert Israel, Aug 07 2014
  • Mathematica
    Table[Binomial[5n+1,n],{n,0,20}]  (* Harvey P. Dale, Jan 23 2011 *)

Formula

a(n) is asymptotic to c*(3125/256)^n/sqrt(n) with c=0.557.... [c = 5^(3/2)/(sqrt(Pi)*2^(7/2)) = 0.55753878629774... - Vaclav Kotesovec, Feb 14 2019 and Aug 20 2025]
8*n*(4*n+1)*(2*n-1)*(4*n-1)*a(n) -5*(5*n+1)*(5*n-3)*(5*n-2)*(5*n-1)*a(n-1)=0. - R. J. Mathar, Jul 17 2014
G.f.: hypergeom([2/5, 3/5, 4/5, 6/5], [1/2, 3/4, 5/4], (3125/256)*x). - Robert Israel, Aug 07 2014
a(n) = [x^n] 1/(1 - x)^(2*(2*n+1)). - Ilya Gutkovskiy, Oct 10 2017
From Seiichi Manyama, Aug 16 2025: (Start)
a(n) = Sum_{k=0..n} binomial(5*n-k,n-k).
G.f.: 1/(1 - x*g^3*(5+g)) where g = 1+x*g^5 is the g.f. of A002294.
G.f.: g^2/(5-4*g) where g = 1+x*g^5 is the g.f. of A002294.
G.f.: B(x)^2/(1 + 4*(B(x)-1)/5), where B(x) is the g.f. of A001449. (End)

A194560 G.f.: Sum_{n>=1} G_n(x)^n where G_n(x) = x + x*G_n(x)^n.

Original entry on oeis.org

1, 2, 2, 4, 2, 10, 2, 20, 14, 49, 2, 217, 2, 438, 310, 1580, 2, 6352, 2, 18062, 7824, 58799, 2, 258971, 2532, 742915, 246794, 2729095, 2, 11154954, 2, 35779660, 8414818, 129644809, 242354, 531132915, 2, 1767263211, 300830821, 6593815523, 2, 26289925026, 2, 91708135773
Offset: 1

Views

Author

Paul D. Hanna, Aug 28 2011

Keywords

Comments

Number of Dyck n-paths with all ascents of equal length. - David Scambler, Nov 17 2011
From Gus Wiseman, Feb 15 2019: (Start)
Also the number of uniform (all blocks have the same size) non-crossing set partitions of {1,...,n}. For example, the a(3) = 2 through a(6) = 10 uniform non-crossing set partitions are:
{{123}} {{1234}} {{12345}} {{123456}}
{{1}{2}{3}} {{12}{34}} {{1}{2}{3}{4}{5}} {{123}{456}}
{{14}{23}} {{126}{345}}
{{1}{2}{3}{4}} {{156}{234}}
{{12}{34}{56}}
{{12}{36}{45}}
{{14}{23}{56}}
{{16}{23}{45}}
{{16}{25}{34}}
{{1}{2}{3}{4}{5}{6}}
(End)

Examples

			G.f.: A(x) = x + 2*x^2 + 2*x^3 + 4*x^4 + 2*x^5 + 10*x^6 + 2*x^7 + ...
where
A(x) = G_1(x) + G_2(x)^2 + G_3(x)^3 + G_4(x)^4 + G_5(x)^5 + ...
and G_n(x) = x + x*G_n(x)^n is given by:
G_n(x) = Sum_{k>=0} C(n*k+1,k)/(n*k+1)*x^(n*k+1),
G_n(x)^n = Sum_{k>=1} C(n*k,k)/(n*k-k+1)*x^(n*k);
the first few expansions of G_n(x)^n begin:
G_1(x) = x + x^2 + x^3 + x^4 + x^5 + ...
G_2(x)^2 = x^2 + 2*x^4 + 5*x^6 + 14*x^8 + ... + A000108(n)*x^(2*n) + ...
G_3(x)^3 = x^3 + 3*x^6 + 12*x^9 + 55*x^12 + ... + A001764(n)*x^(3*n) + ...
G_4(x)^4 = x^4 + 4*x^8 + 22*x^12 + 140*x^16 + ... + A002293(n)*x^(4*n) + ...
G_5(x)^5 = x^5 + 5*x^10 + 35*x^15 + 285*x^20 + ... + A002294(n)*x^(5*n) + ...

Links

Crossrefs

Row sums of A306437.
Cf. A000108, A001263, A001764, A016098, A038041, A061095, A125181, A134264, A194558, A306418, A306438.

Programs

  • Mathematica
    Table[Sum[Binomial[n,d]/(n-d+1),{d,Divisors[n]}],{n,20}] (* Gus Wiseman, Feb 15 2019 *)
  • PARI
    {a(n)=if(n<1,0,sumdiv(n,d,binomial(n,d)/(n-d+1)))}
  • PARI
    {a(n)=polcoeff(sum(m=1,n,serreverse(x/(1+x^m+x*O(x^n)))^m),n)}

Formula

a(n) = Sum_{d|n} C(n,d)/(n-d+1).
G.f.: Sum_{n>=1} Series_Reversion( x/(1+x^n) )^n.

A213225 G.f. satisfies: A(x) = 1/(1 - x/A(-x*A(x)^4)).

Original entry on oeis.org

1, 1, 2, 6, 20, 76, 313, 1375, 6337, 30243, 148129, 739172, 3737993, 19077868, 97955307, 504707999, 2604312205, 13436676965, 69229324721, 355854322633, 1823672937884, 9314227843463, 47406130512872, 240498260267049, 1216833204738419, 6146116088495029, 31030233400282749
Offset: 0

Views

Author

Paul D. Hanna, Jun 06 2012

Keywords

Comments

Compare g.f. to:
(1) G(x) = 1/(1 - x/G(-x*G(x)^3)^1) when G(x) = 1/(1 - x*G(x)^1) (A000108).
(2) G(x) = 1/(1 - x/G(-x*G(x)^5)^2) when G(x) = 1/(1 - x*G(x)^2) (A001764).
(3) G(x) = 1/(1 - x/G(-x*G(x)^7)^3) when G(x) = 1/(1 - x*G(x)^3) (A002293).
(4) G(x) = 1/(1 - x/G(-x*G(x)^9)^4) when G(x) = 1/(1 - x*G(x)^4) (A002294).

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 20*x^4 + 76*x^5 + 313*x^6 +...
Related expansions:
A(x)^4 = 1 + 4*x + 14*x^2 + 52*x^3 + 201*x^4 + 816*x^5 + 3468*x^6 +...
1/A(-x*A(x)^4) = 1 + x + 3*x^2 + 9*x^3 + 35*x^4 + 146*x^5 + 656*x^6 +...

Crossrefs

Cf. A213226, A213227, A213228, A213229, A213230, A213231, A213232, A213233.
Cf. A213091, A213092, A213093, A213094, A213095, A213096, A213098.
Cf. A213099, A213100, A213101, A213102, A213103, A213104, A213105.
Cf. A213108, A213109, A213110, A213111, A213112, A213113.

Programs

  • Mathematica
    terms = 26; A[] = 1; Do[A[x] = 1/(1-x/A[-x*A[x]^4]) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Aug 23 2025 *)
  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1/(1-x/subst(A, x, -x*subst(A^4, x, x+x*O(x^n)))) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

A213226 G.f. satisfies: A(x) = 1/(1 - x/A(-x*A(x)^5)).

Original entry on oeis.org

1, 1, 2, 7, 27, 122, 607, 3208, 17688, 99803, 571238, 3292738, 19001315, 109303307, 624615928, 3537913240, 19843769848, 110273489737, 608712132055, 3355449334452, 18624818099047, 105191779542849, 610586100129734, 3662333209225714, 22652502251884322
Offset: 0

Views

Author

Paul D. Hanna, Jun 06 2012

Keywords

Comments

Compare g.f. to:
(1) G(x) = 1/(1 - x/G(-x*G(x)^3)^1) when G(x) = 1/(1 - x*G(x)^1) (A000108).
(2) G(x) = 1/(1 - x/G(-x*G(x)^5)^2) when G(x) = 1/(1 - x*G(x)^2) (A001764).
(3) G(x) = 1/(1 - x/G(-x*G(x)^7)^3) when G(x) = 1/(1 - x*G(x)^3) (A002293).
(4) G(x) = 1/(1 - x/G(-x*G(x)^9)^4) when G(x) = 1/(1 - x*G(x)^4) (A002294).

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 27*x^4 + 122*x^5 + 607*x^6 +...
Related expansions:
A(x)^5 = 1 + 5*x + 20*x^2 + 85*x^3 + 380*x^4 + 1801*x^5 + 9045*x^6 +...
1/A(-x*A(x)^5) = 1 + x + 4*x^2 + 14*x^3 + 66*x^4 + 336*x^5 + 1805*x^6 +...

Crossrefs

Cf. A213225, A213227, A213228, A213229, A213230, A213231, A213232, A213233.
Cf. A213091, A213092, A213093, A213094, A213095, A213096, A213098.
Cf. A213099, A213100, A213101, A213102, A213103, A213104, A213105.
Cf. A213108, A213109, A213110, A213111, A213112, A213113.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1/(1-x/subst(A, x, -x*subst(A^5, x, x+x*O(x^n)))) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

A213228 G.f. satisfies: A(x) = 1/(1 - x/A(-x*A(x)^6)^2).

Original entry on oeis.org

1, 1, 3, 14, 73, 440, 2862, 19991, 146939, 1125413, 8896018, 72067978, 595097838, 4987609871, 42290465703, 361845473658, 3117830204185, 27009650432888, 234932107635587, 2049479335366836, 17915253987741538, 156799716352350344, 1373180896765862962
Offset: 0

Views

Author

Paul D. Hanna, Jun 06 2012

Keywords

Comments

Compare g.f. to:
(1) G(x) = 1/(1 - x/G(-x*G(x)^3)^1) when G(x) = 1/(1 - x*G(x)^1) (A000108).
(2) G(x) = 1/(1 - x/G(-x*G(x)^5)^2) when G(x) = 1/(1 - x*G(x)^2) (A001764).
(3) G(x) = 1/(1 - x/G(-x*G(x)^7)^3) when G(x) = 1/(1 - x*G(x)^3) (A002293).
(4) G(x) = 1/(1 - x/G(-x*G(x)^9)^4) when G(x) = 1/(1 - x*G(x)^4) (A002294).

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 14*x^3 + 73*x^4 + 440*x^5 + 2862*x^6 +...
Related expansions:
A(x)^6 = 1 + 6*x + 33*x^2 + 194*x^3 + 1188*x^4 + 7656*x^5 + 51583*x^6 +...
1/A(-x*A(x)^6)^2 = 1 + 2*x + 9*x^2 + 44*x^3 + 268*x^4 + 1750*x^5 + 12422*x^6 +...

Crossrefs

Cf. A213225, A213226, A213227, A213229, A213230, A213231, A213232, A213233.
Cf. A213091, A213092, A213093, A213094, A213095, A213096, A213098.
Cf. A213099, A213100, A213101, A213102, A213103, A213104, A213105.
Cf. A213108, A213109, A213110, A213111, A213112, A213113.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1/(1-x/subst(A^2, x, -x*subst(A^6, x, x+x*O(x^n)))) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
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