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 1-10 of 10 results.

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

Views

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

A386380 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/6)} a(6*k) * a(n-1-6*k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 15, 24, 34, 45, 57, 70, 154, 253, 368, 500, 650, 819, 1827, 3045, 4495, 6200, 8184, 10472, 23562, 39627, 59052, 82251, 109668, 141778, 320866, 543004, 814506, 1142295, 1533939, 1997688, 4540200, 7718340, 11633440, 16398200, 22137570
Offset: 0

Views

Author

Seiichi Manyama, Jul 20 2025

Keywords

Links

Crossrefs

Cf. A002296, A233832, A233833, A386392, A233834, A130565.
Cf. A047749, A118968, A124753, A386379, A386396.

Programs

  • Maple
    A386380 := proc(n)
    option remember ;
    if n = 0 then
        1;
    else
        add(procname(6*k)*procname(n-1-6*k),k=0..floor((n-1)/6)) ;
    end if;
end proc:
seq(A386380(n),n=0..80) ; # R. J. Mathar, Jul 30 2025
  • PARI
    apr(n, p, r) = r*binomial(n*p+r, n)/(n*p+r);
a(n) = apr(n\6, 7, n%6+1);

Formula

For k=0..5, a(6*n+k) = (k+1) * binomial(7*n+k+1,n)/(7*n+k+1).
G.f. A(x) satisfies A(x) = 1/(1 - x * Product_{k=0..5} A(w^k*x)), where w = exp(Pi*i/3).

A235536 a(n) = binomial(8*n, 2*n) / (6*n + 1).

Original entry on oeis.org

1, 4, 140, 7084, 420732, 27343888, 1882933364, 134993766600, 9969937491420, 753310723010608, 57956002331347120, 4524678117939182220, 357557785658996609700, 28545588568201512137904, 2298872717007844035521848, 186533392975795702301759056
Offset: 0

Views

Author

Bruno Berselli, Jan 12 2014

Keywords

Comments

This is the case l=6, k=2 of binomial((l+k)*n,k*n)/((l*n+1)/gcd(k,l*n+1)), see Theorem 1.1 in Zhi-Wei Sun's paper.
First bisection of A002293.
Also, the sequence is between A002296 and A235535.

Links

Crossrefs

Cf. similar sequences generated by binomial((l+k)*n,k*n)/(l*n+1), where l is divisible by all the factors of k: A000108 (l=1, k=1), A001764 (l=2, k=1), A002293 (l=3, k=1), A002294 (l=4, k=1), A002295 (l=5, k=1), A002296 (l=6, k=1), A007556 (l=7, k=1), A062994 (l=8, k=1), A059968 (l=9, k=1), A230388 (l=10, k=1), A048990 (l=2, k=2), A235534 (l=4, k=2), this sequence (l=6, k=2), A187357 (l=3, k=3), A235535 (l=6, k=3).

Programs

  • Magma
    l:=6; k:=2; [Binomial((l+k)*n,k*n)/(l*n+1): n in [0..20]]; /* where l is divisible by all the prime factors of k */
  • Mathematica
    Table[Binomial[8 n, 2 n]/(6 n + 1), {n, 0, 20}]

Formula

a(n) = A124753(6*n).
From Ilya Gutkovskiy, Jun 21 2018: (Start)
G.f.: 6F5(1/8,1/4,3/8,5/8,3/4,7/8; 1/3,1/2,2/3,5/6,7/6; 65536*x/729).
a(n) ~ 2^(16*n-1)/(sqrt(Pi)*3^(6*n+3/2)*n^(3/2)). (End)

A124752 Inverse of number triangle A124749.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 2, 2, 2, 2, 1, 0, 0, 0, 3, 3, 3, 3, 2, 1, 0, 0, 0, 4, 4, 4, 4, 3, 2, 1, 0, 0, 0, 9, 9, 9, 9, 7, 5, 3, 1
Offset: 0

Views

Author

Paul Barry, Nov 06 2006

Keywords

Comments

Row sums (and fourth column) are A124753.

Examples

			Triangle begins
1,
0, 1,
0, 0, 1,
0, 0, 0, 1,
0, 0, 0, 1, 1,
0, 0, 0, 1, 1, 1,
0, 0, 0, 1, 1, 1, 1,
0, 0, 0, 2, 2, 2, 2, 1,
0, 0, 0, 3, 3, 3, 3, 2, 1,
0, 0, 0, 4, 4, 4, 4, 3, 2, 1,
0, 0, 0, 9, 9, 9, 9, 7, 5, 3, 1

Crossrefs

Cf. A124747.

A385691 E.g.f. A(x) satisfies A(x) = exp( x*(A(x) + A(w*x) + A(w^2*x))/3 ), where w = exp(2*Pi*i/3).

Original entry on oeis.org

1, 1, 1, 1, 5, 21, 61, 568, 4257, 20917, 286451, 3099141, 21555865, 390273898, 5524889553, 49790422501, 1121734897937, 19631020478229, 217441607213557, 5862333450708460, 122222268766006641, 1606671304363320805, 50443794604147639487, 1220712011020970521461
Offset: 0

Views

Author

Seiichi Manyama, Jul 07 2025

Keywords

Crossrefs

Cf. A058014, A124753.

Programs

  • Mathematica
    terms = 24;  w = Exp[2*Pi*I/3]; A[] = 1; Do[A[x] = Exp[x*(A[x] + A[w*x] + A[w^2*x])/3] + O[x]^terms // Normal, terms]; Simplify[CoefficientList[A[x], x]Range[0,terms-1]!] (* Stefano Spezia, Jul 07 2025 *)

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/3)} (3*k+1) * binomial(n-1,3*k) * a(3*k) * a(n-1-3*k).

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).

A084080 Length of lists created by n substitutions k -> Range[k+1,1,-3] starting with {1}, counting down from k+1 to 1 step -3.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 9, 15, 22, 52, 91, 140, 340, 612, 969, 2394, 4389, 7084, 17710, 32890, 53820, 135720, 254475, 420732, 1068012, 2017356, 3362260, 8579560, 16301164, 27343888, 70068713
Offset: 0

Views

Author

Wouter Meeussen, May 11 2003

Keywords

Comments

Would appear to coincide with row sums of the inverse of the Riordan array (1-x^3,x(1-x^3)). These row sums have g.f. 1/(1-y-y^3+y^4) where y^4-y+x=0. - Paul Barry, May 10 2005

Examples

			{1}, {2}, {3}, {4, 1}, {5, 2, 2}, {6, 3, 3, 3}, {7, 4, 1, 4, 1, 4, 1, 4, 1}

Crossrefs

Cf. A124753.

Programs

  • Mathematica
    Length/@Flatten/@NestList[ # /. k_Integer:>Range[k+1, 1, -3]&, {1}, 21]

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).

A386202 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/3)} binomial(n-1,3*k) * a(3*k) * a(n-1-3*k).

Original entry on oeis.org

1, 1, 1, 1, 2, 6, 16, 52, 234, 1018, 4724, 27864, 166816, 1018096, 7421220, 56215420, 427276280, 3714931512, 33908654224, 309043657936, 3126424467816, 33317327728936, 354276443249552, 4093007897140128, 49813497858533344, 605442506092221760, 7871720463184084560
Offset: 0

Views

Author

Seiichi Manyama, Jul 14 2025

Keywords

Crossrefs

Cf. A000111, A000142, A386203.
Cf. A124753, A385691.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, (i-1)\3, binomial(i-1, 3*j)*v[3*j+1]*v[i-3*j])); v;

Formula

E.g.f. A(x) satisfies A'(x) = A(x) * (A(x) + A(w*x) + A(w^2*x))/3, where w = exp(2*Pi*i/3).

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).
Showing 1-10 of 10 results.

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