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 1-5 of 5 results.

A193147 Expansion of 1/(1 - x - 2*x^3 - x^5).

Original entry on oeis.org

1, 1, 1, 3, 5, 8, 15, 26, 45, 80, 140, 245, 431, 756, 1326, 2328, 4085, 7168, 12580, 22076, 38740, 67985, 119305, 209365, 367411, 644761, 1131476, 1985603, 3484490, 6114853, 10730820, 18831276, 33046585, 57992715, 101770120, 178594110, 313410816, 549997641
Offset: 0

Views

Author

Johannes W. Meijer, Jul 20 2011

Keywords

Comments

The Ze3 sums, see A180662 for the definition of these sums, of the "Races with Ties" triangle A035317 equal this sequence.
Number of tilings of a 5 X 2n rectangle with 5 X 1 pentominoes. - M. Poyraz Torcuk, Dec 18 2021

Links

Crossrefs

Bisection of A003520.
Cf. A035317, A180662, A373578.

Programs

  • Maple
    A193147 := proc(n) option remember: if n>=-4 and n<=-1 then 0 elif n=0 then 1 else procname(n-1) + 2*procname(n-3) + procname(n-5) fi: end: seq(A193147(n), n=0..32);
  • Mathematica
    Series[1/(1 - x - 2*x^3 - x^5), {x, 0, 32}] // CoefficientList[#, x]& (* Jean-François Alcover, Apr 02 2015 *)
  • Maxima
    a(n):=sum(sum(binomial(j,3*n-5*m+2*j)*binomial(2*m-n,j)*2^(3*n-5*m+2*j), j,0,2*m-n),m,floor((n+1)/2),n); /* Vladimir Kruchinin, Mar 10 2013 */

Formula

G.f.: 1/(1-x-2*x^3-x^5) = -1 / ( (1+x+x^2)*(x^3-x^2+2*x-1) ).
a(n) = a(n-1) + 2*a(n-3) + a(n-5) with a(n) = 0 for n= -4, -3, -2, -1 and a(0) = 1.
a(n) = (5*b(n+1) - 4*b(n) + 3*b(n-1) + 2*c(n) + 3*c(n-1))/7 with b(n) = A005314(n) and c(n) = A049347(n).
G.f.: 1 + x/(U(0)-x) where G(k)= 1 - x^2*(k+1)/(1 - 1/(1 + (k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 17 2012
a(n) = Sum_{m=floor((n+1)/2)..n} Sum_{j=0..2*m-n} C(j,3*n-5*m+2*j) * C(2*m-n,j) * 2^(3*n-5*m+2*j). - Vladimir Kruchinin, Mar 10 2013
With offset 1, the INVERT transform of (1 + 2x^2 + x^4). - Gary W. Adamson, Mar 30 2017
a(n) = Sum_{k=0..floor(2*n/5)} binomial(2*n-4*k,k). - Seiichi Manyama, Jun 14 2024

A373620 Expansion of e.g.f. exp(x / (1 - x^2)^2).

Original entry on oeis.org

1, 1, 1, 13, 49, 481, 3841, 38221, 464353, 5368609, 82042561, 1151767981, 20242097041, 342921513793, 6705416722369, 133590317946541, 2880298682358721, 65597610230669761, 1556262483879791233, 39569880403136366029, 1030778206965403668721
Offset: 0

Views

Author

Seiichi Manyama, Jun 11 2024

Keywords

Crossrefs

Cf. A012150, A088009, A373619.
Cf. A082579, A373578.

Programs

  • Maple
    A373620 := proc(n)
    add(binomial(2*n-3*k-1,k)/(n-2*k)!,k=0..floor(n/2)) ;
    %*n! ;
end proc:
seq(A373620(n),n=0..80) ; # R. J. Mathar, Jun 11 2024
  • PARI
    a(n) = n!*sum(k=0, n\2, binomial(2*n-3*k-1, k)/(n-2*k)!);

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} binomial(2*n-3*k-1,k)/(n-2*k)!.
a(n) == 1 mod 12.
a(n) ~ 2^(-1/6) * 3^(-1/2) * exp(1/48 + 2^(-5/3)*n^(1/3) + 3*2^(-4/3)*n^(2/3) - n) * n^(n - 1/6). - Vaclav Kotesovec, Jun 11 2024
D-finite with recurrence a(n) -a(n-1) -3*(n-1)*(n-2)*a(n-2) -3*(n-1)*(n-2)*a(n-3) +3*(n-1)*(n-2)*(n-3)*(n-4)*a(n-4) -(n-1)*(n-2)*(n-3)*(n-4)*(n-5)*(n-6)*a(n-6)=0. - R. J. Mathar, Jun 11 2024

A373577 Expansion of e.g.f. exp(x * (1 + x^2)^(3/2)).

Original entry on oeis.org

1, 1, 1, 10, 37, 136, 1261, 6616, 45865, 479872, 3206521, 31165696, 356045581, 3082798720, 37528974757, 443190912256, 4792765859281, 69943918698496, 875123733523825, 11059833224507392, 179428023035501941, 2557848382674927616, 37699048392962570461
Offset: 0

Views

Author

Seiichi Manyama, Jun 10 2024

Keywords

Crossrefs

Cf. A118395, A190863, A373578.

Programs

  • PARI
    a(n) = n!*sum(k=0, n\2, binomial(3*n/2-3*k, k)/(n-2*k)!);

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} binomial(3*n/2-3*k,k)/(n-2*k)!.
a(n) == 1 mod 9.

A373708 Expansion of e.g.f. exp(x * (1 + x^4)^2).

Original entry on oeis.org

1, 1, 1, 1, 1, 241, 1441, 5041, 13441, 393121, 10946881, 99902881, 559025281, 2335441681, 182348406241, 4382526067921, 48882114328321, 355837396998721, 5157802930734721, 312898934463543361, 7129755898022511361, 89524038506304371761, 773103613914955683361
Offset: 0

Views

Author

Seiichi Manyama, Jun 14 2024

Keywords

Crossrefs

Cf. A361278, A373578, A373707.
Cf. A190877, A373524, A373525, A373526.
Cf. A373706.

Programs

  • PARI
    a(n) = n!*sum(k=0, 2*n\9, binomial(2*n-8*k, k)/(n-4*k)!);

Formula

a(n) = n! * Sum_{k=0..floor(2*n/9)} binomial(2*n-8*k,k)/(n-4*k)!.
a(n) == 1 (mod 240).
a(n) = a(n-1) + 10*(n-1)*(n-2)*(n-3)*(n-4)*a(n-5) + 9*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)*(n-6)*(n-7)*(n-8)*a(n-9).

A373707 Expansion of e.g.f. exp(x * (1 + x^3)^2).

Original entry on oeis.org

1, 1, 1, 1, 49, 241, 721, 6721, 124321, 913249, 4243681, 94818241, 1640604241, 14642181841, 131026944049, 3669304504321, 62536989802561, 627395160826561, 10818406189690561, 308036857749752449, 5219006583104930161, 65146235714284117681
Offset: 0

Views

Author

Seiichi Manyama, Jun 14 2024

Keywords

Crossrefs

Cf. A361278, A373578, A373708.
Cf. A190875, A373522, A373523.
Cf. A373680.

Programs

  • PARI
    a(n) = n!*sum(k=0, 2*n\7, binomial(2*n-6*k, k)/(n-3*k)!);

Formula

a(n) = n! * Sum_{k=0..floor(2*n/7)} binomial(2*n-6*k,k)/(n-3*k)!.
a(n) == 1 (mod 48).
a(n) = a(n-1) + 8*(n-1)*(n-2)*(n-3)*a(n-4) + 7*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)*(n-6)*a(n-7).
Showing 1-5 of 5 results.

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