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 1-4 of 4 results.

A119363 a(n) = Sum_{k=0..n} C(n,3k)^2.

Original entry on oeis.org

1, 1, 1, 2, 17, 101, 402, 1275, 3921, 14114, 58601, 243695, 950578, 3537847, 13166791, 50514102, 198627921, 782913717, 3054480306, 11824753551, 45823049817, 178682390994, 700285942731, 2747647985241, 10767833451954, 42164261091351, 165225573240651
Offset: 0

Views

Author

Paul Barry, May 16 2006

Keywords

Comments

a(n) - A119364(n) = A119365(n).

Links

Crossrefs

Central coefficients of number triangle A119335.
a(n) = A119335(2n, n).
Cf. A024493, A119358, A139468, A139469.

Programs

  • Mathematica
    Table[Sum[Binomial[n,3k]^2, {k,0,n}], {n,0,30}] (* Vaclav Kotesovec, Mar 12 2019 *)
Table[HypergeometricPFQ[{1/3 - n/3, 1/3 - n/3, 2/3 - n/3, 2/3 - n/3, -n/3, -n/3}, {1/3, 1/3, 2/3, 2/3, 1}, 1], {n, 0, 30}] (* Vaclav Kotesovec, Mar 12 2019 *)

Formula

From Vaclav Kotesovec, Mar 12 2019: (Start)
Recurrence: (n-2)*(n-1)*n*(637*n^6 - 11466*n^5 + 84364*n^4 - 324394*n^3 + 686227*n^2 - 755060*n + 336132)*a(n) = 3*(n-2)*(n-1)*(1274*n^7 - 23569*n^6 + 180194*n^5 - 733383*n^4 + 1699606*n^3 - 2208294*n^2 + 1449504*n - 351000)*a(n-1) - 3*(n-2)*(3185*n^8 - 63700*n^7 + 539028*n^6 - 2512118*n^5 + 7020469*n^4 - 11971242*n^3 + 12050010*n^2 - 6446736*n + 1362744)*a(n-2) + (14014*n^9 - 315315*n^8 + 3072678*n^7 - 16986046*n^6 + 58535088*n^5 - 129861691*n^4 + 184326992*n^3 - 159830656*n^2 + 75517728*n - 14313456)*a(n-3) + 3*(n-3)*(3185*n^8 - 63700*n^7 + 538391*n^6 - 2501394*n^5 + 6946794*n^4 - 11707256*n^3 + 11530544*n^2 - 5915328*n + 1142208)*a(n-4) + 18*(n-4)*(n-3)*(2*n - 9)*(637*n^6 - 7644*n^5 + 36589*n^4 - 88858*n^3 + 114124*n^2 - 71840*n + 16440)*a(n-5).
a(n) ~ 4^n / (3*sqrt(Pi*n)). (End)

Extensions

Edited by N. J. A. Sloane, Jun 12 2008

A119366 Number of rooted planar n-trees whose number of leaves is equal to 1 modulo 3.

Original entry on oeis.org

0, 1, 1, 1, 2, 11, 51, 177, 519, 1513, 5042, 18866, 71270, 257974, 905425, 3193737, 11578842, 42930441, 159998493, 593445318, 2194106568, 8138471667, 30399156174, 114219616809, 430344635913, 1622777285682, 6125465491551
Offset: 0

Views

Author

Paul Barry, May 16 2006

Keywords

Comments

a(n)+A119365(n)+A119367(n)=A000108(n).

Programs

  • Maple
    A119366 := proc(n)
    if n = 0 then
        0;
    else
        add(binomial(n,3*k+1)*binomial(n,3*k),k=0..n/3) ;
        %/n ;
    end if;
end proc: # R. J. Mathar, Dec 02 2014

Formula

a(n)=sum{k=0..n, if(mod(n-k,3)=1, (1/n)*C(n,k)*C(n,k+1), 0)}
a(0)=0, a(n)=sum{k=0..floor(n/3), (1/n)*C(n,3k+1)C(n,3k)},n>0; - Paul Barry, Jan 25 2007
Conjecture D-finite with recurrence +n*(881*n-4580)*(n-2)*(n+1)*a(n) -3*n*(612*n^3-2827*n^2-2988*n+10135)*a(n-1) +3*(-3088*n^4+42803*n^3-190361*n^2+313702*n-167988)*a(n-2) +(43042*n^4-600920*n^3+2924411*n^2-5860777*n+4115562)*a(n-3) +3*(-38600*n^4+558681*n^3-2904370*n^2+6389913*n-4965528)*a(n-4) +3*(-14776*n^4+162695*n^3-434711*n^2-415064*n+1878084)*a(n-5) -9*(n-6)*(10835*n^3-106831*n^2+290611*n-173519)*a(n-6) +54*(n-6)*(n-7)*(593*n-1429)*(2*n-13)*a(n-7)=0. - R. J. Mathar, Feb 03 2025

A119367 Number of rooted planar n-trees whose number of leaves is equal to 2 modulo 3.

Original entry on oeis.org

0, 0, 1, 3, 6, 11, 30, 126, 519, 1836, 5877, 18866, 65472, 242463, 905425, 3307371, 11889414, 42930441, 157641714, 586908936, 2194106568, 8189323686, 30541703733, 114219616809, 429214875498, 1619584557885, 6125465491551
Offset: 0

Views

Author

Paul Barry, May 16 2006

Keywords

Comments

a(n)+A119365(n)+A119366(n)=A000108(n).

Formula

a(n)=sum{k=0..n, if(mod(n-k,3)=2, (1/n)*C(n,k)*C(n,k+1), 0)}.

A119364 Central coefficients of number triangle A119335.

Original entry on oeis.org

0, 1, 1, 1, 11, 81, 351, 1149, 3529, 12601, 52724, 222641, 879308, 3295384, 12303201, 47320365, 186738507, 739129809, 2894481813, 11237844615, 43647142533, 170543919327, 669744238998, 2633027605209, 10337488816041, 40544676533466
Offset: 0

Views

Author

Paul Barry, May 16 2006

Keywords

Comments

A119363(n)-a(n)=A119365(n).

Formula

a(n)=sum{k=0..n-1, C(n+1,3k)*C(n-1,3k)}; a(n)=A119335(2n,n+1).
Showing 1-4 of 4 results.

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

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