A118971 -id:A118971 - cp's OEIS frontend

A366328 G.f. satisfies A(x) = (1 + x) * (1 + x/A(x)^4).

1, 2, -7, 60, -612, 6898, -82806, 1038076, -13431940, 178040315, -2405137161, 32992706368, -458336721104, 6435090557964, -91167680664004, 1301665779507128, -18710805300530504, 270559054510943509, -3932893180646204203, 57437414168562779574, -842365843304975785062

Offset: 0

Seiichi Manyama, Oct 07 2023

sign

Cf. A073157, A364336, A364337, A364338, A364339, A366325, A366326, A366327.

Cf. A118971, A364408.

a(n) = (-1)^(n-1)*sum(k=0, n, binomial(5*k-1, k)*binomial(n+3*k-2, n-k)/(5*k-1));

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

A381746 Expansion of exp( Sum_{k>=1} binomial(10k-1,2k) * x^k/k ).

Original entry on oeis.org

1, 36, 2586, 235884, 24284907, 2689924444, 312907382800, 37699275223260, 4663450108073401, 588854988193808392, 75589352418472567340, 9834912295258236849604, 1294095251234713917535805, 171909332777340128148714400, 23024035140764003881788203616

Offset: 0

Seiichi Manyama, Mar 05 2025

Seiichi Manyama, Table of n, a(n) for n = 0..462

Cf. A079489, A381744, A381745.

Cf. A118971, A381752.

my(N=20, x='x+O('x^N)); Vec(exp(sum(k=1, N, binomial(10*k-1, 2*k)*x^k/k)))

G.f. A(x) satisfies A(x^2) = B(x) * B(-x), where B(x) is the g.f. of A118971.
a(n) = Sum_{k=0..2*n} (-1)^k * A118971(k) * A118971(2*n-k).
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(10*k-1,2*k) * a(n-k).
G.f.: B(x)^4, where B(x) is the g.f. of A381752.

A380553 G.f. A(x) satisfies x = Sum_{n>=1} A( x^n(1-x)^(4n) ).

Original entry on oeis.org

1, 3, 25, 200, 1770, 16351, 158223, 1577328, 16112031, 167708890, 1772645419, 18974340640, 205263418940, 2240623110285, 24648785800540, 272994642782048, 3041495503591364, 34064252952038769, 383302465665133013, 4331178750570145160, 49126274119206904221, 559128033687856289017

Offset: 1

Paul D. Hanna, Feb 16 2025

nonn

Moebius transform of A118971.

			G.f.: A(x) = x + 3*x^2 + 25*x^3 + 200*x^4 + 1770*x^5 + 16351*x^6 + 158223*x^7 + 1577328*x^8 + 16112031*x^9 + 167708890*x^10 + ...
where x = Sum_{n>=1} A( x^n*(1-x)^(4*n) ).
RELATED SERIES.
Sum_{n>=1} a(n) * x^n/(1-x^n) = x + 4*x^2 + 26*x^3 + 204*x^4 + 1771*x^5 + 16380*x^6 + 158224*x^7 + 1577532*x^8 + ... + A118971(n)*x^(n) + ...
which equals x*F(x)^4 where F(x) = 1 + x*F(x)^5 is the g.f. of A002294.

Paul D. Hanna, Table of n, a(n) for n = 1..500

Cf. A346936, A034742, A380551, A380552, A118971, A002294, A008683.

\\ As the Moebius transform of A118971 \\
{a(n) = sumdiv(n,d, moebius(n/d) * binomial(5*d-1,d-1)*4/(5*d-1) )}
for(n=1,30,print1(a(n),", "))

\\ By definition x = Sum_{n>=1} A( x^n*(1-x)^(4*n) ) \\
{a(n) = my(V=[0,1]); for(i=0,n, V = concat(V,0); A = Ser(V);
V[#V] = polcoef(x - sum(m=1,#V, subst(A,x, x^m*(1-x)^(4*m) +x*O(x^#V)) ),#V-1)); V[n+1]}
for(n=1,30,print1(a(n),", "))

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) x = Sum_{n>=1} A( x^n*(1-x)^(4*n) ).
(2) x = Sum_{n>=1} a(n) * x^n*(1-x)^(4*n) / (1 - x^n*(1-x)^(4*n)).
(3) x*F(x)^4 = Sum_{n>=1} a(n) * x^n/(1-x^n) where F(x) = 1 + x*F(x)^5 is the g.f. of A002294.
(4) a(n) = Sum_{d|n} mu(n/d) * binomial(5*d-1,d-1)*4/(5*d-1), where mu is the Moebius function A008683.

A162382 Triangle, read by rows, defined by: T(n,k) = 1/((k+1)n-1) binomial((k+1)n-1,n) for n,k>0.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 5, 7, 3, 1, 14, 30, 15, 4, 1, 42, 143, 91, 26, 5, 1, 132, 728, 612, 204, 40, 6, 1, 429, 3876, 4389, 1771, 385, 57, 7, 1, 1430, 21318, 32890, 16380, 4095, 650, 77, 8, 1, 4862, 120175, 254475, 158224, 46376, 8184, 1015, 100, 9, 1, 16796, 690690, 2017356

Offset: 1

Aminul Huq (aminul(AT)brandeis.edu), Jul 02 2009

T(n,k) counts number of lattice paths with steps (1,k) and (1,-1) starting at the origin and ending at height 1 with i vertices on or below the x-axis for i=1,2,...,(r+1)n-1. For k=1, T(n,1) are the Catalan numbers A000108, k=2 gives the sequence A006013, k=3 gives the sequence A006632, k=4 gives the sequence A118971, etc.

Cf. A000108, A006013, A006632, A118971, etc.

TableForm[ Table[1/((k + 1) n - 1) Binomial[(k + 1) n - 1, n], {k, 1, 10}, {n, 1, 10}]]

Satisfies xf^k(x)=1-f^{-1}(x). Can also be written as T(n,k) = 1/n binomial((k+1)n-2,n-1) = 1/(kn-1) binomial((k+1)n-2,n)

A370844 Expansion of (1/x) * Series_Reversion( x / (1/(1-x)^4 + x) ).

Original entry on oeis.org

1, 5, 35, 295, 2760, 27556, 287564, 3098780, 34216020, 385106280, 4401850866, 50957904938, 596231618166, 7039674475190, 83767631913840, 1003564049999916, 12094813260406732, 146534778450346908, 1783695235540931924, 21803615393276536720, 267537602528379374851

Offset: 0

Seiichi Manyama, Mar 03 2024

nonn

Index entries for reversions of series

Cf. A007317, A369616, A369617.

Cf. A118971.

my(N=30, x='x+O('x^N)); Vec(serreverse(x/(1/(1-x)^4+x))/x)

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

a(n) = Sum_{k=0..n} binomial(n,k) * A118971(k).

a(n) = hypergeom([4/5, 6/5, 7/5, 8/5, -n], [5/4, 3/2, 7/4, 2], -3125/256). - Stefano Spezia, Mar 03 2024

A349023 G.f. satisfies: A(x) = 1/(1 - x/(1 - x*A(x))^4)^2.

Original entry on oeis.org

1, 2, 11, 64, 417, 2892, 20941, 156500, 1198049, 9346690, 74042938, 594001236, 4815995027, 39399831458, 324840184326, 2696343599336, 22514057175337, 188977375146888, 1593661234493561, 13495942411592260, 114723671513478118, 978570384358686064

Offset: 0

Seiichi Manyama, Nov 06 2021

nonn

Cf. A118971, A321798, A349024.

a(n, s=4, t=2) = sum(k=0, n, binomial(t*n-(t-1)*(k-1), k)*binomial(n+(s-1)*k-1, n-k)/(n-k+1));

If g.f. satisfies: A(x) = 1/(1 - x/(1 - x*A(x))^s)^t, then a(n) = Sum_{k=0..n} binomial(t*n-(t-1)*(k-1),k) * binomial(n+(s-1)*k-1,n-k)/(n-k+1).

A349024 G.f. satisfies: A(x) = 1/(1 - x/(1 - x*A(x))^4)^3.

Original entry on oeis.org

1, 3, 18, 124, 951, 7764, 66200, 582594, 5252133, 48254668, 450186720, 4253328540, 40612877001, 391300954065, 3799506069816, 37142836241690, 365255937037437, 3610755090793272, 35861607622930556, 357670540310182842, 3580797575489620740

Offset: 0

Seiichi Manyama, Nov 06 2021

nonn

Cf. A118971, A321798, A349023.

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

If g.f. satisfies: A(x) = 1/(1 - x/(1 - x*A(x))^s)^t, then a(n) = Sum_{k=0..n} binomial(t*n-(t-1)*(k-1),k) * binomial(n+(s-1)*k-1,n-k)/(n-k+1).

A379187 G.f. A(x) satisfies A(x) = 1/((1 - xA(x)^2) (1 - x*A(x))^3).

Original entry on oeis.org

1, 4, 30, 286, 3091, 36063, 442898, 5642628, 73893561, 988585443, 13453580815, 185661101085, 2592069904059, 36545520229810, 519601325300487, 7441580996167052, 107255985242888943, 1554576968046707916, 22644622298400113411, 331322620547205661043

Offset: 0

Seiichi Manyama, Dec 17 2024

nonn

Cf. A118971, A369617, A379188.

Cf. A369215.

a(n) = sum(k=0, n, binomial(n+2*k+1, k)*binomial(4*n+2*k+2, n-k)/(n+2*k+1));

a(n) = Sum_{k=0..n} binomial(n+2*k+1,k) * binomial(4*n+2*k+2,n-k)/(n+2*k+1).

A386392 a(n) = 4 * binomial(7n+4,n)/(7n+4).

Original entry on oeis.org

1, 4, 34, 368, 4495, 59052, 814506, 11633440, 170574723, 2552698720, 38832808586, 598724403680, 9335085772194, 146936230074004, 2331703871687400, 37263447339612480, 599206511767593099, 9688121925389895636, 157401957319775436400, 2568427016865897264000

Offset: 0

Seiichi Manyama, Jul 20 2025

nonn

Wikipedia, Fuss-Catalan number

Cf. A118971, A212073, A234463, A234507, A234527, A234870.

Cf. A002296, A386380.

apr(n, p, r) = r*binomial(n*p+r, n)/(n*p+r);
a(n) = apr(n, 7, 4);

a(n) = r * binomial(n*p+r,n)/(n*p+r), the Fuss-Catalan number with p=7 and r=4.
a(n) = A386380(6*n+3).
G.f. A(x) satisfies A(x) = (1 + x*A(x)^(p/r))^r, where p=7, r=4.
G.f.: B(x)^4, where B(x) is the g.f. of A002296.

A386558 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = binomial((k+1)*n+k-1,n)/(n+1).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 7, 5, 0, 1, 4, 15, 30, 14, 0, 1, 5, 26, 91, 143, 42, 0, 1, 6, 40, 204, 612, 728, 132, 0, 1, 7, 57, 385, 1771, 4389, 3876, 429, 0, 1, 8, 77, 650, 4095, 16380, 32890, 21318, 1430, 0, 1, 9, 100, 1015, 8184, 46376, 158224, 254475, 120175, 4862, 0

Offset: 0

Seiichi Manyama, Jul 26 2025

			Square array begins:
  1,   1,    1,     1,      1,      1,       1, ...
  0,   1,    2,     3,      4,      5,       6, ...
  0,   2,    7,    15,     26,     40,      57, ...
  0,   5,   30,    91,    204,    385,     650, ...
  0,  14,  143,   612,   1771,   4095,    8184, ...
  0,  42,  728,  4389,  16380,  46376,  109668, ...
  0, 132, 3876, 32890, 158224, 548340, 1533939, ...

Wikipedia, Fuss-Catalan number

Columns k=0..10 give A000007, A000108, A006013, A006632, A118971, A130564(n+1), A130565(n+1), A234466, A234513, A234573, A235340.
Main diagonal gives A177784(n+1).
Cf. A162382.

a(n, k) = binomial((k+1)*n+k-1, n)/(n+1);

For k > 0, A(n,k) = r * binomial(n*p+r,n)/(n*p+r), the Fuss-Catalan number with p=k+1 and r=k.

G.f. of column k: (1/x) Series_Reverion( x*(1-x)^k ).

cp's OEIS Frontend

A366328 G.f. satisfies A(x) = (1 + x) * (1 + x/A(x)^4).

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A381746 Expansion of exp( Sum_{k>=1} binomial(10*k-1,2*k) * x^k/k ).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A380553 G.f. A(x) satisfies x = Sum_{n>=1} A( x^n*(1-x)^(4*n) ).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

A162382 Triangle, read by rows, defined by: T(n,k) = 1/((k+1)n-1) binomial((k+1)n-1,n) for n,k>0.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A370844 Expansion of (1/x) * Series_Reversion( x / (1/(1-x)^4 + x) ).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Formula

A349023 G.f. satisfies: A(x) = 1/(1 - x/(1 - x*A(x))^4)^2.

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A349024 G.f. satisfies: A(x) = 1/(1 - x/(1 - x*A(x))^4)^3.

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A379187 G.f. A(x) satisfies A(x) = 1/((1 - x*A(x)^2) * (1 - x*A(x))^3).

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A386392 a(n) = 4 * binomial(7*n+4,n)/(7*n+4).

Views

Author

Keywords

Links

Crossrefs

Programs

A381746 Expansion of exp( Sum_{k>=1} binomial(10k-1,2k) * x^k/k ).

A380553 G.f. A(x) satisfies x = Sum_{n>=1} A( x^n(1-x)^(4n) ).

A379187 G.f. A(x) satisfies A(x) = 1/((1 - xA(x)^2) (1 - x*A(x))^3).

A386392 a(n) = 4 * binomial(7n+4,n)/(7n+4).