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.

A119365 Generalized Catalan numbers for triangle A119335.

Original entry on oeis.org

1, 0, 0, 1, 6, 20, 51, 126, 392, 1513, 5877, 21054, 71270, 242463, 863590, 3193737, 11889414, 43783908, 159998493, 586908936, 2175907284, 8138471667, 30541703733, 114620380032, 430344635913, 1619584557885, 6116422089050
Offset: 0

Views

Author

Paul Barry, May 16 2006

Keywords

Comments

Counts rooted planar n-trees whose number of leaves is divisible by 3.

Crossrefs

Cf. A000108, A001263, A119335, A119366, A119367.

Programs

  • Maple
    A119365 := proc(n)
    local k;
    if n = 0 then
        return 1
    end if;
    a := 0 ;
    for k from 0 to n do
        if modp(n-k,3) = 0 then
            a := a+binomial(n,k)*binomial(n,k+1) ;
        end if;
    end do:
    a/n;
end proc:
seq(A119365(n),n=0..40) ; # R. J. Mathar, Oct 30 2014
  • Mathematica
    A119335[n_, k_] := Sum[Binomial[k, 3j] Binomial[n-k, 3j], {j, 0, n-k}];
a[n_] := A119335[2n, n] - A119335[2n, n+1];
Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Sep 14 2023 *)

Formula

a(n) = A119335(2n,n) - A119335(2n,n+1).
a(n) = Sum_{k=0..n} if(mod(n-k,3)=0, (1/n)*C(n,k)*C(n,k+1), 0).
a(n) + A119366(n) + A119367(n) = A000108(n).

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).

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

A119336 Expansion of (1-x)^4/((1-x)^6 - x^6).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 16, 45, 130, 341, 804, 1730, 3460, 6555, 12016, 21845, 40410, 77540, 155080, 320001, 669526, 1398101, 2884776, 5858126, 11716252, 23166783, 45536404, 89478485, 176565486, 350739488, 701478976, 1410132405, 2841788170
Offset: 0

Views

Author

Paul Barry, May 14 2006

Keywords

Comments

Row sums of A119335. Binomial transform of (1+x)/(1-x)^6.
Equals binomial transform of [1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, ...]. - Gary W. Adamson, Mar 14 2009

Links

Crossrefs

Cf. A119335, A306847.

Programs

  • Mathematica
    CoefficientList[Series[(1-x)^4/((1-x)^6-x^6),{x,0,40}],x] (* or *) LinearRecurrence[{6,-15,20,-15,6},{1,2,3,4,5},40] (* Harvey P. Dale, Dec 25 2015 *)
  • PARI
    {a(n) = sum(k=0, n\6, binomial(n+1, 6*k+1))} \\ Seiichi Manyama, Mar 22 2019

Formula

a(n) = Sum_{k=0..n} Sum_{j=0..n-k} C(k,3j)*C(n-k,3j).
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5), with a(0)=1, a(1)=2, a(2)=3, a(3)=4, a(4)=5. - Harvey P. Dale, Dec 25 2015
a(n) = Sum_{k=0..floor(n/6)} binomial(n+1,6*k+1). - Seiichi Manyama, Mar 22 2019

A307090 Number triangle T(n,k) = Sum_{j=0..n-k} (-1)^j * binomial(k,2*j) * binomial(n-k,2*j).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, -2, -2, 1, 1, 1, 1, -5, -8, -5, 1, 1, 1, 1, -9, -17, -17, -9, 1, 1, 1, 1, -14, -29, -34, -29, -14, 1, 1, 1, 1, -20, -44, -54, -54, -44, -20, 1, 1, 1, 1, -27, -62, -74, -74, -74, -62, -27, 1, 1, 1, 1, -35, -83, -90, -74, -74, -90, -83, -35, 1, 1
Offset: 0

Views

Author

Seiichi Manyama, Mar 24 2019

Keywords

Examples

			Triangle begins:
n\k | 0  1    2    3    4    5    6  7  8
----+-------------------------------------
0   | 1;
1   | 1, 1;
2   | 1, 1,   1;
3   | 1, 1,   1,   1;
4   | 1, 1,   0,   1,   1;
5   | 1, 1,  -2,  -2,   1,   1;
6   | 1, 1,  -5,  -8,  -5,   1,   1;
7   | 1, 1,  -9, -17, -17,  -9,   1, 1;
8   | 1, 1, -14, -29, -34, -29, -14, 1, 1;

Links

Crossrefs

Row sums give A099587(n+1).
T(2*n,n) gives A307091.
Cf. A007318, A098593, A119326, A119335.

Programs

  • Mathematica
    T[n_, k_] := Sum[(-1)^j * Binomial[k, 2*j] * Binomial[n - k, 2*j], {j, 0, n - k}]; Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Amiram Eldar, May 20 2021 *)

A307156 Number triangle T(n,k) = Sum_{j=0..n-k} (-1)^j * binomial(k,3*j) * binomial(n-k,3*j).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, -3, -3, 1, 1, 1, 1, 1, 1, -9, -15, -9, 1, 1, 1, 1, 1, 1, -19, -39, -39, -19, 1, 1, 1, 1, 1, 1, -34, -79, -99, -79, -34, 1, 1, 1, 1, 1, 1, -55, -139, -199, -199, -139, -55, 1, 1, 1
Offset: 0

Views

Author

Seiichi Manyama, Mar 27 2019

Keywords

Examples

			Triangle begins:
n\k | 0  1  2   3    4   5  6  7  8
----+-------------------------------
0   | 1;
1   | 1, 1;
2   | 1, 1, 1;
3   | 1, 1, 1,  1;
4   | 1, 1, 1,  1,   1;
5   | 1, 1, 1,  1,   1,  1;
6   | 1, 1, 1,  0,   1,  1, 1;
7   | 1, 1, 1, -3,  -3,  1, 1, 1;
8   | 1, 1, 1, -9, -15, -9, 1, 1, 1;

Links

Crossrefs

Row sums give A307089.
T(2*n,n) gives A307158.
Cf. A119326, A119335, A307090.

Programs

  • Mathematica
    T[n_, k_] := Sum[(-1)^j * Binomial[k, 3*j] * Binomial[n - k, 3*j], {j, 0, n - k}]; Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Amiram Eldar, May 20 2021 *)

A119337 Number triangle T(n,k)=sum{i=0..n, (-1)^(n-i)*C(n,i)*sum{j=0..i-k, C(k,3j)*C(i-k,3j)}}.

Original entry on oeis.org

1, 0, 1, 0, -1, 1, 0, 1, -2, 1, 0, -1, 3, -3, 1, 0, 1, -4, 6, -4, 1, 0, -1, 5, -9, 10, -5, 1, 0, 1, -6, 12, -16, 15, -6, 1, 0, -1, 7, -15, 19, -25, 21, -7, 1, 0, 1, -8, 18, -16, 20, -36, 28, -8, 1, 0, -1, 9, -21, 4, 24, 6, -49, 36, -9, 1
Offset: 0

Views

Author

Paul Barry, May 14 2006

Keywords

Comments

Row sums have g.f. (1+x)/(1-x)^6. Multiply by Pascal's triangle A007318 to get A119335.

Examples

			Triangle begins
1,
0, 1,
0, -1, 1,
0, 1, -2, 1,
0, -1, 3, -3, 1,
0, 1, -4, 6, -4, 1,
0, -1, 5, -9, 10, -5, 1,
0, 1, -6, 12, -16, 15, -6, 1,
0, -1, 7, -15, 19, -25, 21, -7, 1,
0, 1, -8, 18, -16, 20, -36, 28, -8, 1,
0, -1, 9, -21, 4, 24, 6, -49, 36, -9, 1

Formula

Column k has g.f. (x/(1+x))^k*sum{j=0..k, C(k,3j)x^(3j)}
Showing 1-7 of 7 results.

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

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