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.

Previous Showing 41-49 of 49 results.

A386379 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/5)} a(5*k) * a(n-1-5*k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 13, 21, 30, 40, 51, 114, 190, 280, 385, 506, 1150, 1950, 2925, 4095, 5481, 12586, 21576, 32736, 46376, 62832, 145299, 250971, 383838, 548340, 749398, 1741844, 3025308, 4654320, 6690585, 9203634, 21475146, 37456650, 57887550
Offset: 0

Views

Author

Seiichi Manyama, Jul 20 2025

Keywords

Links

Crossrefs

Cf. A002295, A212071, A212072, A212073, A130564.
Cf. A047749, A118968, A124753, A386380.

Programs

  • PARI
    apr(n, p, r) = r*binomial(n*p+r, n)/(n*p+r);
a(n) = apr(n\5, 6, n%5+1);

Formula

For k=0..4, a(5*n+k) = (k+1) * binomial(6*n+k+1,n)/(6*n+k+1).
G.f. A(x) satisfies A(x) = 1/(1 - x * Product_{k=0..4} A(w^k*x)), where w = exp(2*Pi*i/5).

A385618 G.f. A(x) satisfies A(x) = 1/( 1 - x*(A(2*x) + A(3*x)) ).

Original entry on oeis.org

1, 2, 14, 230, 9014, 913334, 254986934, 203241812630, 471322195238102, 3214892041613961206, 64937611960188470964662, 3901256965326759127330935830, 699101347969640933511109922382422, 374397435055450676411068538643233721206, 599979003238812649083869782544110463986119734
Offset: 0

Views

Author

Seiichi Manyama, Jul 05 2025

Keywords

Crossrefs

Cf. A007689, A015083, A015084, A047749, A385617.

Programs

  • Mathematica
    terms = 15; A[] = 1; Do[A[x] = 1/( 1 - x*(A[2*x] + A[3*x]) ) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Jul 05 2025 *)
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, i-1, (2^j+3^j)*v[j+1]*v[i-j])); v;

Formula

a(0) = 1; a(n) = Sum_{k=0..n-1} (2^k+3^k) * a(k) * a(n-1-k).

A385699 G.f. A(x) satisfies A(x) = 1/( 1 - x*(A(x) + A(-x))*(A(x) + A(w*x) + A(w^2*x))/6 ), where w = exp(2*Pi*i/3).

Original entry on oeis.org

1, 1, 1, 2, 5, 13, 24, 88, 181, 523, 1616, 4891, 10540, 42009, 94953, 294102, 957259, 3028320, 6864540, 28208447, 66180997, 211105506, 703497178, 2273009790, 5283518340, 22058432677, 52795736539, 171169636087, 578132050147, 1891182035377, 4462525373212
Offset: 0

Views

Author

Seiichi Manyama, Jul 07 2025

Keywords

Crossrefs

Cf. A047749, A124753, A217138, A385698.

Formula

a(0) = 1; a(n) = Sum_{i, j, k>=0 and i+2*j+3*k=n-1} a(i) * a(2*j) * a(3*k).

A386396 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/7)} a(7*k) * a(n-1-7*k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 17, 27, 38, 50, 63, 77, 92, 200, 325, 468, 630, 812, 1015, 1240, 2728, 4488, 6545, 8925, 11655, 14763, 18278, 40508, 67158, 98728, 135751, 178794, 228459, 285384, 635628, 1059380, 1566040, 2165800, 2869685, 3689595, 4638348
Offset: 0

Views

Author

Seiichi Manyama, Jul 20 2025

Keywords

Links

Crossrefs

Cf. A007556, A234461, A234462, A234463, A234464, A234465, A234466.
Cf. A047749, A118968, A124753, A386379, A386380.

Programs

  • PARI
    apr(n, p, r) = r*binomial(n*p+r, n)/(n*p+r);
a(n) = apr(n\7, 8, n%7+1);

Formula

For k=0..6, a(7*n+k) = (k+1) * binomial(8*n+k+1,n)/(8*n+k+1).
G.f. A(x) satisfies A(x) = 1/(1 - x * Product_{k=0..6} A(w^k*x)), where w = exp(2*Pi*i/7).

A123922 Number of 2143-avoiding Dumont paths of the 2nd kind of length 2n.

Original entry on oeis.org

1, 1, 2, 6, 21, 84, 360, 1650, 7865, 39039, 198744, 1039584, 5534928, 30046752, 165257136, 922280634, 5199131025, 29644168125, 170375955750, 988180543350, 5768664340725, 33927954699600, 200617471267200, 1193673954039840
Offset: 0

Views

Author

R. J. Mathar, Nov 20 2006

Keywords

Examples

			For n=2, there are 3 Dumont permutations of the 2nd kind of length 2n=4, namely {2143,3142,4132}.
Avoiding 2143, the cardinality of this set is reduced to a(2)=2.

Links

Crossrefs

Cf. A001469, A047749.

Programs

  • Mathematica
    b[n_] := If[EvenQ[n], Binomial[3n/2, n/2]/(n+1), Binomial[(3n-1)/2, (n+1)/2 ]/n];
a[n_] := b[n] b[n+1];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Jul 27 2018 *)
  • PARI
    A047749(n)={ my(m=floor(n/2)); if(n % 2, binomial(3*m+1,m+1)/(2*m+1), binomial(3*m,m)/(2*m+1)); }
a(n)={ A047749(n)*A047749(n+1); }

Formula

a(n) = A047749(n)*A047749(n+1).
Conjecture: 16*n*(n+2)*(n+1)^2*a(n) -108*n*(n+1)*(2*n-1)*a(n-1) -9*(3*n-5)*(3*n-1)*(3*n-4)*(3*n-2)*a(n-2)=0. - R. J. Mathar, Jan 25 2013

A134565 Expansion of reversion of (x - 2*x^2) / (1 - x)^3.

Original entry on oeis.org

1, -1, 2, -3, 7, -12, 30, -55, 143, -273, 728, -1428, 3876, -7752, 21318, -43263, 120175, -246675, 690690, -1430715, 4032015, -8414640, 23841480, -50067108, 142498692, -300830572, 859515920, -1822766520, 5225264024, -11124755664, 31983672534, -68328754959
Offset: 1

Views

Author

Michael Somos, Nov 01 2007

Keywords

Examples

			G.f. = x - x^2 + 2*x^3 - 3*x^4 + 7*x^5 - 12*x^6 + 30*x^7 - 55*x^8 + 143*x^9 + ...

Links

Crossrefs

Cf. A001764, A005161, A006013, A047749, A080956, A099576.

Programs

  • Mathematica
    a[ n_] := With[ {m = Quotient[n, 2]}, If[n < 1, 0, -(-1)^n Binomial[n + m, n - m] / (2 m + 1)]]; (* Michael Somos, Oct 16 2015 *)
a[ n_] := If[n < 1, 0, SeriesCoefficient[ InverseSeries[ Series[(x - 2 x^2) / (1 - x)^3, {x, 0, n}]], {x, 0, n}]]; (* Michael Somos, Oct 16 2015 *)
a[n_] := (-1)^(n-1)*Binomial[2*n, n-1]*Hypergeometric2F1[-n+1, n, -2*n, -1] / n; Flatten[Table[a[n], {n, 1, 32}]] (* Detlef Meya, Dec 26 2023 *)
  • PARI
    {a(n) = my( m = n\2); if( n<1, 0, -(-1)^n * binomial( n + m, n - m) / (2 * m + 1))};
  • PARI
    {a(n) = if( n<1, 0, polcoeff( serreverse( (x - 2 * x^2) / (1 - x)^3 + x * O(x^n) ), n))};
  • PARI
    {a(n) = if( n<1, 0, polcoeff( 1 / ( 1 + 1 / serreverse( x - x^3 + x * O(x^n) )), n))};

Formula

Given g.f. A(x), then 1 = (1/A(x) + 1/A(-x)) / 2.
a(n) = -(-1)^n * binomial(n + m, n - m) / (2*m + 1) where m = floor(n/2) if n>0.
From Michael Somos, Apr 13 2012 (Start)
a(n) = -(-1)^n * A047749(n) unless n=0. a(2*n) = - A001764(n) unless n=0. a(2*n + 1) = A006013(n).
Reversion of A080956 with offset 1.
Hankel transform is A005161 omitting first 1.
n * a(n) = -(-1)^n * A099576(n-1). (End)
D-finite with recurrence +8*n*(n+1)*a(n) -36*n*(n-2)*a(n-1) +6*(-9*n^2+18*n-14)*a(n-2) +27*(3*n-7)*(3*n-8)*a(n-3)=0. - R. J. Mathar, Sep 24 2021
a(n) = (-1)^(n-1)*binomial(2*n, n-1)*hypergeom([-n+1, n], [-2*n], -1) / n. - Detlef Meya, Dec 26 2023

A172026 Riordan array (f(x^2), x*f(x^2)) where f(x) is the g.f. of A001764.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 3, 0, 3, 0, 1, 0, 7, 0, 4, 0, 1, 12, 0, 12, 0, 5, 0, 1, 0, 30, 0, 18, 0, 6, 0, 1, 55, 0, 55, 0, 25, 0, 7, 0, 1, 0, 143, 0, 88, 0, 33, 0, 8, 0, 1, 273, 0, 273, 0, 130, 0, 42, 0, 9, 0, 1, 0, 728, 0, 455, 0, 182, 0, 52, 0, 10, 0, 1, 1428, 0, 1428, 0, 700, 0, 245, 0
Offset: 0

Views

Author

Philippe Deléham, Jan 23 2010

Keywords

Comments

Another version of A110616. Riordan production matrix is: (x/(1-x^2), 1/(1-x^2)).

Examples

			Triangle begins : 1 ; 0,1 ; 1,0,1 ; 0,2,0,1 ; 3,0,3,0,1 ; 0,7,0,4,0,1 ; 12,0,12,0,5,0,1 ; ...

Crossrefs

Cf. A047749, A092276, A143603

Formula

Sum_{k, 0<=k<=n} T(n,k)= A047749(n+1).

A299293 Number of (n + 1, n + 2)-core partitions into odd parts.

Original entry on oeis.org

1, 2, 4, 7, 17, 31, 80, 152, 404, 790, 2140, 4271, 11729, 23767, 65952, 135221, 378321, 782968, 2205168, 4598804, 13023324, 27332956, 77761008
Offset: 0

Views

Author

N. J. A. Sloane, Feb 14 2018

Keywords

Links

Crossrefs

Cf. A047749.

A331519 a(0) = 1; a(n) = Sum_{k=1..n} (binomial(n,k) mod 2) * a(k-1) * a(n-k).

Original entry on oeis.org

1, 1, 1, 3, 3, 9, 15, 63, 63, 189, 315, 1323, 1701, 6237, 12285, 59535, 59535, 178605, 297675, 1250235, 1607445, 5893965, 11609325, 56260575, 63761985, 213790185, 393824025, 1811590515, 2531725875, 10025634465, 21210236775, 109876902975, 109876902975, 329630708925
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 19 2020

Keywords

Crossrefs

Cf. A000108, A001147, A047749, A047999, A062177, A166966.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[Mod[Binomial[n, k], 2] a[k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 33}]
Previous Showing 41-49 of 49 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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