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 1-7 of 7 results.

A073155 Leftmost column sequence of triangle A073153.

Original entry on oeis.org

1, 1, 4, 14, 56, 237, 1046, 4762, 22198, 105430, 508384, 2482297, 12248416, 60980875, 305955356, 1545397464, 7852100294, 40105277640, 205798130604, 1060467961508, 5485199090812, 28469067353686, 148220323891460
Offset: 0

Views

Author

Paul D. Hanna, Jul 29 2002

Keywords

Examples

			a(3)=a(0)*[a(2)+a(1)]+[a(1)+a(0)]*[a(1)+a(0)]+[a(2)+a(1)]*a(0) =1*[4+1] + [1+1]*[1+1] + [4+1]*1 = 5 + 2*2 + 5 = 14.

Links

Crossrefs

Cf. A073153, A073156, A073157.
Cf. A052709, A360076, A360082, A360083.

Formula

Convolution of sequence formed from sum of adjacent terms yields the original sequence without the first term:
a(n+1) = Sum_{k=0..n} [a(k) + a(k-1)] * [a(n-k) + a(n-k-1)], where a(-1)=0.
G.f.: 1/2*(1-(1-4*x*(1+x)^2)^(1/2))/x/(1+x)^2. - Vladeta Jovovic, Oct 10 2003
a(n) = Sum_{k=0..n} C(2k,n-k)*C(k). - Paul Barry, Jul 09 2006
Conjecture: (n+1)*a(n) + (-3*n+4)*a(n-1) + 2*(-6*n+7)*a(n-2) + 2*(-6*n+11)*a(n-3) + 2*(-2*n+5)*a(n-4)=0. - R. J. Mathar, Nov 26 2012
G.f. A(x) satisfies: A(x) = 1 + x * ((1 + x) * A(x))^2. - Ilya Gutkovskiy, Jul 10 2020

Extensions

More terms from Vladeta Jovovic, Oct 10 2003

A360082 a(n) = Sum_{k=0..n} binomial(4*k,n-k) * Catalan(k).

Original entry on oeis.org

1, 1, 6, 27, 134, 709, 3892, 22004, 127250, 749230, 4476386, 27071344, 165398868, 1019405720, 6330482488, 39571612357, 248796862550, 1572300095758, 9981970108384, 63633339713190, 407162295120570, 2614059813642256, 16834457481559076
Offset: 0

Views

Author

Seiichi Manyama, Jan 25 2023

Keywords

Crossrefs

Cf. A052709, A073155, A360076, A360083.
Cf. A000108, A099235.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(4*k, n-k)*binomial(2*k, k)/(k+1));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(2/(1+sqrt(1-4*x*(1+x)^4)))

Formula

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

A360083 a(n) = Sum_{k=0..n} binomial(5*k,n-k) * Catalan(k).

Original entry on oeis.org

1, 1, 7, 35, 189, 1092, 6538, 40278, 253730, 1626858, 10582616, 69669273, 463319257, 3107941405, 21004392887, 142882885210, 977562617826, 6722361860888, 46438235933700, 322111000796428, 2242538435656450, 15665017062799230, 109761527468995102
Offset: 0

Views

Author

Seiichi Manyama, Jan 25 2023

Keywords

Links

Crossrefs

Cf. A052709, A073155, A360076, A360082.
Cf. A000108, A360090.

Programs

  • Mathematica
    Table[Sum[Binomial[5k,n-k]CatalanNumber[k],{k,0,n}],{n,0,30}] (* Harvey P. Dale, Jul 13 2025 *)
  • PARI
    a(n) = sum(k=0, n, binomial(5*k, n-k)*binomial(2*k, k)/(k+1));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(2/(1+sqrt(1-4*x*(1+x)^5)))

Formula

G.f. A(x) satisfies A(x) = 1/(1 - x * (1+x)^5 * A(x)).
G.f.: 2 / (1 + sqrt( 1 - 4*x*(1+x)^5 )).

A376145 E.g.f. satisfies A(x) = exp( x * (1+x)^3 * A(x) ).

Original entry on oeis.org

1, 1, 9, 88, 1265, 23916, 558427, 15608986, 508516017, 18936594712, 793902926771, 37017671474334, 1900666877186761, 106576903636156084, 6481047448001720427, 424870924596413523106, 29871349825140536394593, 2242231079099137007066544
Offset: 0

Views

Author

Seiichi Manyama, Sep 11 2024

Keywords

Links

Crossrefs

Cf. A362771, A362772, A376146.
Cf. A360076, A367789.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x*(1+x)^3))))

Formula

E.g.f.: exp( -LambertW(-x * (1+x)^3) ).
a(n) = n! * Sum_{k=0..n} (k+1)^(k-1) * binomial(3*k,n-k)/k!.

A382892 G.f. A(x) satisfies A(x) = 1/( 1 - x * (1+x)^3 * A(x) )^3.

Original entry on oeis.org

1, 3, 24, 190, 1659, 15309, 146986, 1453536, 14704917, 151479031, 1583533308, 16756882194, 179149227231, 1932144798513, 20996553430206, 229678298803028, 2527034248221849, 27947027713469307, 310494250880357488, 3463870813896354726, 38787008808135775299
Offset: 0

Views

Author

Seiichi Manyama, Apr 08 2025

Keywords

Crossrefs

Cf. A360076, A382894.
Cf. A366272, A382614.

Programs

  • PARI
    a(n, r=3, s=3, t=4, u=0) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+r));

Formula

G.f. A(x) satisfies A(x) = ( 1 + x * (1+x)^3 * A(x)^(4/3) )^3.
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(s*k,n-k)/(t*k+u*(n-k)+r).
G.f.: B(x)^3, where B(x) is the g.f. of A366272.

A382894 G.f. A(x) satisfies A(x) = 1/( 1 - x * (1+x)^3 * A(x) )^2.

Original entry on oeis.org

1, 2, 13, 78, 520, 3664, 26859, 202808, 1566693, 12323982, 98381841, 795023284, 6490951398, 53462144788, 443683640945, 3706539244272, 31144893093298, 263052053436600, 2231992880546400, 19016760502183968, 162629329186013523, 1395500273826639540
Offset: 0

Views

Author

Seiichi Manyama, Apr 08 2025

Keywords

Crossrefs

Cf. A360076, A382892.
Cf. A366200, A382613.

Programs

  • PARI
    a(n, r=2, s=3, t=3, u=0) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+r));

Formula

G.f. A(x) satisfies A(x) = ( 1 + x * (1+x)^3 * A(x)^(3/2) )^2.
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(s*k,n-k)/(t*k+u*(n-k)+r).
G.f.: B(x)^2, where B(x) is the g.f. of A366200.

A367285 G.f. satisfies A(x) = 1 + x*A(x)^2 * (1 + x*A(x)^2)^3.

Original entry on oeis.org

1, 1, 5, 26, 159, 1042, 7185, 51340, 376806, 2823734, 21516113, 166196703, 1298413089, 10241803340, 81454834164, 652465062453, 5259084437170, 42624217133130, 347160390473763, 2839928983316595, 23323730673818467, 192237734035157372, 1589602164422747636
Offset: 0

Views

Author

Seiichi Manyama, Nov 12 2023

Keywords

Crossrefs

Cf. A360076, A361305.
Cf. A137953, A367261.
Cf. A367240.

Programs

  • PARI
    a(n, s=3, t=2, u=2) = sum(k=0, n, binomial(t*k+u*(n-k)+1, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+1));

Formula

If g.f. satisfies A(x) = 1 + x*A(x)^t * (1 + x*A(x)^u)^s, then a(n) = Sum_{k=0..n} binomial(t*k+u*(n-k)+1,k) * binomial(s*k,n-k) / (t*k+u*(n-k)+1).
Showing 1-7 of 7 results.

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