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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A078971 Numbers n such that C(4n,n)/(3n+1) (A002293) is not divisible by 4.

Original entry on oeis.org

0, 1, 3, 5, 11, 13, 21, 43, 45, 53, 85, 171, 173, 181, 213, 341, 683, 685, 693, 725, 853, 1365, 2731, 2733, 2741, 2773, 2901, 3413, 5461, 10923, 10925, 10933, 10965, 11093, 11605, 13653, 21845, 43691, 43693, 43701, 43733, 43861, 44373, 46421, 54613
Offset: 1

Views

Author

Benoit Cloitre, Jan 14 2003

Keywords

Comments

Stanica observes that the sequence in binary forms a pattern where 1 bits are inserted into the word 1010101...:
1 11
101 1011 1101
10101 101011 101101 110101
1010101 10101011 10101101 10110101 11010101...

Links

Crossrefs

Cf. A000225 (C(2n, n)/(n+1) is not divisible by 2), A003462 (C(3n, n)/(2n+1) is not divisible by 3), A003463 (C(5n, n)/(4n+1) is not divisible by 5).

Programs

  • Magma
    [n: n in [0..2*10^4] | not IsZero(Binomial(4*n,n) div (3*n+1) mod 4)]; // Vincenzo Librandi, Apr 16 2015
  • Mathematica
    Select[ Range[0, 65000], Mod[ Binomial[4#, # ]/(3# + 1), 4] != 0 &] (* Robert G. Wilson v, Oct 12 2005 *)
  • PARI
    isok(n) = binomial(4*n,n)/(3*n+1) % 4; \\ Michel Marcus, Apr 16 2015
  • Python
    from _future_ import division
A078971_list = []
for t in range(100):
    A078971_list.append((2**(2*t)-1)//3)
    for j in range(t):
        A078971_list.append((2**(2*t+1)+2**(2*j+1)-1)//3) # Chai Wah Wu, Mar 06 2016

Extensions

Comments and more terms from Ralf Stephan, Oct 30 2003
a(28)-a(44) from Robert G. Wilson v, Oct 12 2005

A251574 E.g.f.: exp(4*x*G(x)^3) / G(x)^3 where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 4, 40, 712, 18784, 663424, 29480896, 1581976960, 99585422848, 7198258087936, 587699970912256, 53497834761985024, 5372784803063664640, 590164397145095421952, 70386834555048578596864, 9058611906733586004803584, 1251310862246447324484468736, 184665445630564847038730076160
Offset: 0

Views

Author

Paul D. Hanna, Dec 06 2014

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 40*x^3/3! + 712*x^4/4! + 18784*x^5/5! +...
such that A(x) = exp(4*x*G(x)^3) / G(x)^3
where G(x) = 1 + x*G(x)^4 is the g.f. of A002293:
G(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + 7084*x^6 +...
Note that
A'(x) = exp(4*x*G(x)^3) = 1 + 4*x + 40*x^2/2! + 712*x^3/3! + 18784*x^4/4! +...
LOGARITHMIC DERIVATIVE.
The logarithm of the e.g.f. begins:
log(A(x)) = x + 3*x^2/2 + 15*x^3/3 + 91*x^4/4 + 612*x^5/5 +...
and so A'(x)/A(x) = G(x)^3.
TABLE OF POWERS OF E.G.F.
Form a table of coefficients of x^k/k! in A(x)^n as follows.
n=1: [1, 1,  4,   40,   712,  18784,   663424,   29480896, ...];
n=2: [1, 2, 10,  104,  1840,  47888,  1669696,   73399040, ...];
n=3: [1, 3, 18,  198,  3528,  91152,  3146256,  136990656, ...];
n=4: [1, 4, 28,  328,  5944, 153376,  5257024,  227057728, ...];
n=5: [1, 5, 40,  500,  9280, 240440,  8209600,  352337600, ...];
n=6: [1, 6, 54,  720, 13752, 359424, 12263184,  523933056, ...];
n=7: [1, 7, 70,  994, 19600, 518728, 17737216,  755807920, ...];
n=8: [1, 8, 88, 1328, 27088, 728192, 25020736, 1065353216, ...]; ...
in which the main diagonal begins (see A251584):
[1, 2, 18, 328, 9280, 359424, 17737216, 1065353216, ...]
and is given by the formula:
[x^n/n!] A(x)^(n+1) = 4^(n-2) * (n+1)^(n-3) * (3*n^2 + 13*n + 16) for n>=0.

Crossrefs

Cf. A251584, A251664, A002293, A006632.
Cf. Variants: A243953, A251573, A251575, A251576, A251577, A251578, A251579, A251580.

Programs

  • Mathematica
    Flatten[{1,1,Table[Sum[4^k * n!/k! * Binomial[4*n-k-4, n-k] * (k-1)/(n-1),{k,0,n}],{n,2,20}]}] (* Vaclav Kotesovec, Dec 07 2014 *)
  • PARI
    {a(n) = local(G=1);for(i=1,n,G=1+x*G^4 +x*O(x^n)); n!*polcoeff(exp(4*x*G^3)/G^3, n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {a(n) = if(n==0||n==1, 1, sum(k=0, n, 4^k * n!/k! * binomial(4*n-k-4,n-k) * (k-1)/(n-1) ))}
for(n=0, 20, print1(a(n), ", "))

Formula

Let G(x) = 1 + x*G(x)^4 be the g.f. of A002293, then the e.g.f. A(x) of this sequence satisfies:
(1) A'(x)/A(x) = G(x)^3.
(2) A'(x) = exp(4*x*G(x)^3).
(3) A(x) = exp( Integral G(x)^3 dx ).
(4) A(x) = exp( Sum_{n>=1} A006632(n)*x^n/n ), where A006632(n) = binomial(4*n-2,n)/(3*n-1).
(5) A(x) = F(x/A(x)) where F(x) is the e.g.f. of A251584.
(6) A(x) = Sum_{n>=0} A251584(n)*(x/A(x))^n/n! and
(7) [x^n/n!] A(x)^(n+1) = (n+1)*A251584(n),
where A251584(n) = 4^(n-2) * (n+1)^(n-4) * (3*n^2 + 13*n + 16).
a(n) = Sum_{k=0..n} 4^k * n!/k! * binomial(4*n-k-4, n-k) * (k-1)/(n-1) for n>1.
Recurrence: 3*(3*n-5)*(3*n-4)*(4*n^2 - 23*n + 34)*a(n) = 8*(128*n^5 - 1440*n^4 + 6520*n^3 - 14906*n^2 + 17289*n - 8190)*a(n-1) + 256*(4*n^2 - 15*n + 15)*a(n-2). - Vaclav Kotesovec, Dec 07 2014
a(n) ~ 2^(8*n-9) * n^(n-2) / (3^(3*n-7/2) * exp(n-1)). - Vaclav Kotesovec, Dec 07 2014

A147855 G.f.: 1 / (1 + 4*x*G(x)^2 - 7*x*G(x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 3, 22, 174, 1444, 12323, 107104, 942952, 8381596, 75053100, 676017962, 6118171326, 55591175956, 506805088026, 4633571685968, 42468065811884, 390071875757852, 3589637747968964, 33089300640166360, 305476314574338648, 2823932709938708824, 26137341654281261347
Offset: 0

Views

Author

Paul D. Hanna, Jun 16 2013

Keywords

Examples

			G.f.: A(x) = 1 + 3*x + 22*x^2 + 174*x^3 + 1444*x^4 + 12323*x^5 +...
A related series is G(x) = 1 + x*G(x)^4, where
G(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + 7084*x^6 +...
G(x)^2 = 1 + 2*x + 9*x^2 + 52*x^3 + 340*x^4 + 2394*x^5 + 17710*x^6 +...
G(x)^3 = 1 + 3*x + 15*x^2 + 91*x^3 + 612*x^4 + 4389*x^5 + 32890*x^6 +...
such that A(x) = 1/(1 + 4*x*G(x)^2 - 7*x*G(x)^3).

Links

Crossrefs

Cf. A226705, A002293.
Cf. A026641, A183160, A371753.
Cf. A005810, A078995, A226733, A226761, A385605, A386811.

Programs

  • Mathematica
    Table[Sum[Binomial[2*n+k,n-k]*Binomial[2*n-k,k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jun 16 2013 *)
  • PARI
    {a(n)=sum(k=0, n, binomial(2*n+k, n-k)*binomial(2*n-k, k))}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n)=sum(k=0, n, binomial(k, n-k)*binomial(4*n-k, k))}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n)=local(G=1+x); for(i=0, n,G=1+x*G^4+x*O(x^n)); polcoeff(1/(1+4*x*G^2-7*x*G^3), n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n)=local(G=1+x); for(i=0, n,G=1+x*G^4+x*O(x^n)); polcoeff(1/(1-3*x*G^2-7*x^2*G^6), n)}
for(n=0, 30, print1(a(n), ", "))

Formula

a(n) = Sum_{k=0..n} C(k, n-k) * C(4*n-k, k).
a(n) = Sum_{k=0..n} C(n+k, n-k) * C(3*n-k, k).
a(n) = Sum_{k=0..n} C(2*n+k, n-k) * C(2*n-k, k).
a(n) = Sum_{k=0..n} C(3*n+k, n-k) * C(n-k, k).
a(n) = Sum_{k=0..n} C(4*n+k, n-k) * C(-k, k).
G.f.: 1 / (1 - 3*x*G(x)^2 - 7*x^2*G(x)^6) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.
a(n) ~ 2^(8*n+5/2)/(5*sqrt(Pi*n)*3^(3*n+1/2)). - Vaclav Kotesovec, Jun 16 2013
From Seiichi Manyama, Apr 05 2024: (Start)
a(n) = Sum_{k=0..floor(n/2)} binomial(4*n-2*k-1,n-2*k).
a(n) = [x^n] 1/((1-x^2) * (1-x)^(3*n)). (End)
From Seiichi Manyama, Aug 05 2025: (Start)
a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(4*n+1,k).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(3*n+k,k). (End)
From Seiichi Manyama, Aug 14 2025: (Start)
a(n) = Sum_{k=0..n} (-1)^k * 2^(n-k) * binomial(4*n+1,k) * binomial(4*n-k,n-k).
G.f.: G(x)^2/((-1+2*G(x)) * (4-3*G(x))) where G(x) = 1+x*G(x)^4 is the g.f. of A002293. (End)
G.f.: B(x)^2/(1 + 5*(B(x)-1)/4), where B(x) is the g.f. of A005810. - Seiichi Manyama, Aug 15 2025

A213336 G.f. satisfies A(x) = G(x/(1-x)^4) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 8, 64, 568, 5440, 54888, 574848, 6190872, 68132224, 762874568, 8663106496, 99536424952, 1155012037824, 13516570396968, 159340702404352, 1890451582396632, 22555522916988672, 270466907608087944, 3257754635421506368, 39397587357527547320
Offset: 0

Views

Author

Paul D. Hanna, Jun 09 2012

Keywords

Examples

			G.f.: A(x) = 1 + x + 8*x^2 + 64*x^3 + 568*x^4 + 5440*x^5 + 54888*x^6 +...
G.f.: A(x) = G(x/(1-x)^4) where G(x) = 1 + x*G(x)^4 is g.f. of A002293:
G(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + 7084*x^6 +...

Links

Crossrefs

Cf. A213335, A002293; variants: A006319, A213282.
Partial sums give A349310. - Seiichi Manyama, Oct 03 2023

Programs

  • PARI
    /* G.f. A(x) = G(x/(1-x)^4) where G(x) = 1 + x*G(x)^4: */
{a(n)=local(A, G=1+x); for(i=1, n, G=1+x*G^4+x*O(x^n)); A=subst(G, x, x/(1-x+x*O(x^n))^4); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    /* G.f. A(x) = F(x*A(x)^4) where F(x) = 1 + x/F(-x)^4: */
{a(n)=local(F=1+x+x*O(x^n),A=1); for(i=1, n+1, F=1+x/subst(F^4, x, -x+x*O(x^n))); A=(serreverse(x/F^4)/x)^(1/4);polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Formula

G.f. satisfies: A(x) = F(x*A(x)^4) where F(x) = 1 + x/F(-x)^4 is the g.f. of A213335.
G.f. A(x) satisfies: A(1 - G(-x)) = G(x) = 1 + x*G(x)^4 is the g.f. of A002293.
a(n) = Sum_{k=0..n} binomial(n+3*k-1,n-k) * binomial(4*k,k)/(3*k+1). - Seiichi Manyama, Oct 03 2023

A251664 E.g.f.: exp(4*x*G(x)^3) / G(x) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 3, 26, 430, 10872, 373664, 16295152, 862486944, 53729041280, 3851892172288, 312411790027776, 28284076403710208, 2827642792215049216, 309396856974126428160, 36777992050266076762112, 4719560392385576181243904, 650284066459536965937364992, 95752333835299098922624876544, 15005473998204120386383308390400
Offset: 0

Views

Author

Paul D. Hanna, Dec 07 2014

Keywords

Examples

			E.g.f.: A(x) = 1 + 3*x + 26*x^2/2! + 430*x^3/3! + 10872*x^4/4! + 373664*x^5/5! +...
such that A(x) = exp(4*x*G(x)^3) / G(x)
where G(x) = 1 + x*G(x)^4 is the g.f. of A002293:
G(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + 7084*x^6 +...

Crossrefs

Cf. A251574, A251694, A002293.
Cf. Variants: A243953, A251663, A251665, A251666, A251667, A251668, A251669, A251670.

Programs

  • Mathematica
    Table[Sum[4^k * n!/k! * Binomial[4*n-k-2,n-k] * (3*k-1)/(3*n-1),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Dec 07 2014 *)
  • PARI
    {a(n)=local(G=1); for(i=0, n, G=1+x*G^4 +x*O(x^n)); n!*polcoeff(exp(4*x*G^3)/G, n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {a(n) = sum(k=0, n, 4^k * n!/k! * binomial(4*n-k-2,n-k) * (3*k-1)/(3*n-1) )}
for(n=0, 20, print1(a(n), ", "))

Formula

Let G(x) = 1 + x*G(x)^4 be the g.f. of A002293, then the e.g.f. A(x) of this sequence satisfies:
(1) A'(x)/A(x) = G(x)^3 + 2*G'(x)/G(x).
(2) A(x) = F(x/A(x)^3) where F(x) is the e.g.f. of A251694.
(3) A(x) = Sum_{n>=0} A251694(n)*(x/A(x)^3)^n/n! where A251694(n) = (2*n+1) * (3*n+1)^(n-2) * 4^n.
(4) [x^n/n!] A(x)^(3*n+1) = (2*n+1) * (3*n+1)^(n-1) * 4^n.
a(n) = Sum_{k=0..n} 4^k * n!/k! * binomial(4*n-k-2,n-k) * (3*k-1)/(3*n-1) for n>=0.
Recurrence: 3*(3*n-2)*(3*n-1)*(64*n^3 - 344*n^2 + 598*n - 315)*a(n) = 8*(2048*n^6 - 16128*n^5 + 51136*n^4 - 82160*n^3 + 67332*n^2 - 22212*n - 855)*a(n-1) + 256*(64*n^3 - 152*n^2 + 102*n + 3)*a(n-2). - Vaclav Kotesovec, Dec 07 2014
a(n) ~ 2^(8*n-2) / 3^(3*n-1/2) * n^(n-1) / exp(n-1). - Vaclav Kotesovec, Dec 07 2014

A226733 G.f.: 1 / (1 + 8*x*G(x)^2 - 10*x*G(x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 2, 18, 142, 1186, 10152, 88414, 779508, 6936066, 62159224, 560238728, 5072970366, 46114086446, 420558296888, 3846232573236, 35261290343112, 323952686556354, 2981787128165592, 27491128592627800, 253835886034173848, 2346892194318851016, 21724880414632781472
Offset: 0

Views

Author

Paul D. Hanna, Jun 16 2013

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 18*x^2 + 142*x^3 + 1186*x^4 + 10152*x^5 +...
A related series is G(x) = 1 + x*G(x)^4, where
G(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + 7084*x^6 +...
G(x)^2 = 1 + 2*x + 9*x^2 + 52*x^3 + 340*x^4 + 2394*x^5 + 17710*x^6 +...
G(x)^3 = 1 + 3*x + 15*x^2 + 91*x^3 + 612*x^4 + 4389*x^5 + 32890*x^6 +...
such that A(x) = 1/(1 + 8*x*G(x)^2 - 10*x*G(x)^3).

Links

Crossrefs

Cf. A147855, A183160, A226705, A002293.
Cf. A005810, A078995, A147855, A226761, A385605, A386811.

Programs

  • Mathematica
    Table[Sum[Binomial[2*n+2*k,n-k]*Binomial[2*n-2*k,k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jun 16 2013 *)
  • PARI
    {a(n)=local(G=1+x); for(i=0, n,G=1+x*G^4+x*O(x^n)); polcoeff(1/(1+8*x*G^2-10*x*G^3), n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n)=local(G=1+x); for(i=0, n,G=1+x*G^4+x*O(x^n)); polcoeff(1/(1-2*x*G^2-10*x^2*G^6), n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n)=sum(k=0, n, binomial(2*n+2*k, n-k)*binomial(2*n-2*k, k))}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n)=sum(k=0, n, binomial(2*k, n-k)*binomial(4*n-2*k, k))}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n)=sum(k=0, n, binomial(4*n+2*k, n-k)*binomial(-2*k, k))}
for(n=0, 30, print1(a(n), ", "))

Formula

a(n) = Sum_{k=0..n} C(2*k, n-k) * C(4*n-2*k, k).
a(n) = Sum_{k=0..n} C(n+2*k, n-k) * C(3*n-2*k, k).
a(n) = Sum_{k=0..n} C(2*n+2*k, n-k) * C(2*n-2*k, k).
a(n) = Sum_{k=0..n} C(3*n+2*k, n-k) * C(n-2*k, k).
a(n) = Sum_{k=0..n} C(4*n+2*k, n-k) * C(-2*k, k).
G.f.: 1 / (1 - 2*x*G(x)^2 - 10*x^2*G(x)^6) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.
a(n) ~ 2^(8*n+3/2)/(3^(3*n+3/2)*sqrt(Pi*n)). - Vaclav Kotesovec, Jun 16 2013
From Seiichi Manyama, Aug 05 2025: (Start)
a(n) = [x^n] 1/((1+2*x) * (1-x)^(3*n+1)).
a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(4*n+1,k).
a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(3*n+k,k). (End)
From Seiichi Manyama, Aug 14 2025: (Start)
a(n) = Sum_{k=0..n} (-2)^k * 3^(n-k) * binomial(4*n+1,k) * binomial(4*n-k,n-k).
G.f.: G(x)^2/((-2+3*G(x)) * (4-3*G(x))) where G(x) = 1+x*G(x)^4 is the g.f. of A002293. (End)
G.f.: B(x)^2/(1 + 3*(B(x)-1)/2), where B(x) is the g.f. of A005810. - Seiichi Manyama, Aug 15 2025

A380515 Expansion of e.g.f. exp(x*G(x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 7, 109, 2689, 91261, 3950191, 208064137, 12917499169, 923765042809, 74780847503191, 6760168138392901, 675023676995501857, 73787463232202560309, 8763902701210982610559, 1123850728979698205132641, 154757223522414820829369281, 22775744033825102490806751217
Offset: 0

Views

Author

Seiichi Manyama, Jan 26 2025

Keywords

Crossrefs

Cf. A380513, A380514, A380516.
Cf. A080893, A380511.
Cf. A091695, A250917, A380512.
Cf. A002293, A006632, A370057, A382059.

Programs

  • PARI
    a(n) = if(n==0, 1, 3*n!*sum(k=0, n-1, binomial(3*n+k, k)/((3*n+k)*(n-k-1)!)));

Formula

a(n) = 3 * n! * Sum_{k=0..n-1} binomial(3*n+k,k)/((3*n+k) * (n-k-1)!) for n > 0.
a(n) = U(1-n, 2-4*n, 1), where U is the Tricomi confluent hypergeometric function. - Stefano Spezia, Jan 26 2025
E.g.f.: exp( Series_Reversion( x*(1-x)^3 ) ). - Seiichi Manyama, Mar 15 2025

A380516 Expansion of e.g.f. exp(x*G(x)^4) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 9, 157, 4129, 146001, 6502681, 349790029, 22069858497, 1598577634369, 130757736096361, 11922399644742621, 1199121973234651489, 131887738425602277457, 15748194681225620534649, 2028885239529647188594381, 280525944581514367875035521, 41434950383158772951280658689
Offset: 0

Views

Author

Seiichi Manyama, Jan 26 2025

Keywords

Links

Crossrefs

Cf. A002293, A380513, A380514, A380515.
Cf. A000262, A251568, A380512.
Cf. A382034.

Programs

  • Mathematica
    Join[{1}, Table[(n-1)! * LaguerreL[n-1, 3*n+1, -1], {n, 1, 20}]] (* Vaclav Kotesovec, Jan 26 2025 *)
  • PARI
    a(n) = if(n==0, 1, (n-1)!*pollaguerre(n-1, 3*n+1, -1));

Formula

E.g.f.: exp(G(x)-1), where G(x) is described above.
a(n) = (n-1)! * Sum_{k=0..n-1} binomial(4*n,k)/(n-k-1)! for n > 0.
a(n+1) = n! * LaguerreL(n, 3*n+4, -1).
a(n) = (-1)^(n+1)*U(1-n, 2+3*n, -1), where U is the Tricomi confluent hypergeometric function. - Stefano Spezia, Jan 26 2025
a(n) ~ 2^(8*n + 1) * n^(n-1) / (exp(n - 1/3) * 3^(3*n + 3/2)). - Vaclav Kotesovec, Jan 26 2025
E.g.f.: exp( Series_Reversion( x/(1+x)^4 ) ). - Seiichi Manyama, Mar 15 2025

A226761 G.f.: 1 / (1 + 12*x*G(x)^2 - 13*x*G(x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 16, 118, 1004, 8601, 75076, 662796, 5903676, 52949332, 477533356, 4326309406, 39343725716, 358943047438, 3283745710968, 30112624408488, 276715616909148, 2547523969430508, 23491659440021920, 216942761366305144, 2006084011596742384, 18572529488934397689
Offset: 0

Views

Author

Paul D. Hanna, Jun 16 2013

Keywords

Examples

			G.f.: A(x) = 1 + x + 16*x^2 + 118*x^3 + 1004*x^4 + 8601*x^5 +...
A related series is G(x) = 1 + x*G(x)^4, where
G(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + 7084*x^6 +...
G(x)^2 = 1 + 2*x + 9*x^2 + 52*x^3 + 340*x^4 + 2394*x^5 + 17710*x^6 +...
G(x)^3 = 1 + 3*x + 15*x^2 + 91*x^3 + 612*x^4 + 4389*x^5 + 32890*x^6 +...
such that A(x) = 1/(1 + 12*x*G(x)^2 - 13*x*G(x)^3).

Links

Crossrefs

Cf. A002293.
Cf. A005810, A078995, A147855, A226733, A385605, A386811.

Programs

  • Mathematica
    Table[Sum[Binomial[2*n+3*k,n-k]*Binomial[2*n-3*k,k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jun 17 2013 *)
  • PARI
    {a(n)=local(G=1+x); for(i=0, n, G=1+x*G^4+x*O(x^n)); polcoeff(1/(1+12*x*G^2-13*x*G^3), n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n)=local(G=1+x); for(i=0, n, G=1+x*G^4+x*O(x^n)); polcoeff(1/(1-x*G^2-13*x^2*G^6), n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n)=sum(k=0, n, binomial(2*n+3*k, n-k)*binomial(2*n-3*k, k))}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n)=sum(k=0, n, binomial(3*k, n-k)*binomial(4*n-3*k, k))}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n)=sum(k=0, n, binomial(4*n+3*k, n-k)*binomial(-3*k, k))}
for(n=0, 30, print1(a(n), ", "))

Formula

a(n) = Sum_{k=0..n} C(3*k, n-k) * C(4*n-3*k, k).
a(n) = Sum_{k=0..n} C(n+3*k, n-k) * C(3*n-3*k, k).
a(n) = Sum_{k=0..n} C(2*n+3*k, n-k) * C(2*n-3*k, k).
a(n) = Sum_{k=0..n} C(3*n+3*k, n-k) * C(n-3*k, k).
a(n) = Sum_{k=0..n} C(4*n+3*k, n-k) * C(-3*k, k).
G.f.: 1 / (1 - x*G(x)^2 - 13*x^2*G(x)^6) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.
a(n) ~ 2^(8*n+5/2)/(7*3^(3*n+1/2)*sqrt(Pi*n)). - Vaclav Kotesovec, Jun 17 2013
From Seiichi Manyama, Aug 05 2025: (Start)
a(n) = [x^n] 1/((1+3*x) * (1-x)^(3*n+1)).
a(n) = Sum_{k=0..n} (-4)^(n-k) * binomial(4*n+1,k).
a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(3*n+k,k). (End)
From Seiichi Manyama, Aug 14 2025: (Start)
a(n) = Sum_{k=0..n} (-3)^k * 4^(n-k) * binomial(4*n+1,k) * binomial(4*n-k,n-k).
G.f.: G(x)^2/((-3+4*G(x)) * (4-3*G(x))) where G(x) = 1+x*G(x)^4 is the g.f. of A002293. (End)
G.f.: B(x)^2/(1 + 7*(B(x)-1)/4), where B(x) is the g.f. of A005810. - Seiichi Manyama, Aug 15 2025

A380513 Expansion of e.g.f. exp(x*G(x)) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 3, 31, 649, 20241, 831691, 42281023, 2558247441, 179401012129, 14301145772371, 1276863732880671, 126200478678828313, 13677209933635675441, 1612657716714084149019, 205505541279096688937791, 28144314031348292162103841, 4122178445898981809990411073, 642961375302043479923591655331
Offset: 0

Views

Author

Seiichi Manyama, Jan 26 2025

Keywords

Crossrefs

Cf. A002293, A380514, A380515, A380516.
Cf. A000262, A080893, A251569.

Programs

  • PARI
    a(n) = if(n==0, 1, n!*sum(k=0, n-1, binomial(n+3*k, k)/((n+3*k)*(n-k-1)!)));

Formula

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