A130458 -id:A130458 - cp's OEIS frontend

A069270 Third level generalization of Catalan triangle (0th level is Pascal's triangle A007318; first level is Catalan triangle A009766; 2nd level is A069269).

1, 1, 1, 1, 2, 4, 1, 3, 9, 22, 1, 4, 15, 52, 140, 1, 5, 22, 91, 340, 969, 1, 6, 30, 140, 612, 2394, 7084, 1, 7, 39, 200, 969, 4389, 17710, 53820, 1, 8, 49, 272, 1425, 7084, 32890, 135720, 420732, 1, 9, 60, 357, 1995, 10626, 53820, 254475, 1068012, 3362260

Offset: 0

Henry Bottomley, Mar 12 2002

For the m-th level generalization of Catalan triangle T(n,k) = C(n+mk,k)*(n-k+1)/(n+(m-1)k+1); for n >= k+m: T(n,k) = T(n-m+1,k+1) - T(n-m,k+1); and T(n,n) = T(n+m-1,n-1) = C((m+1)n,n)/(mn+1).

Antidiagonals of convolution matrix of Table 1.5, p. 397, of Hoggatt and Bicknell. - Tom Copeland, Dec 25 2019

			Rows start
  1;
  1,   1;
  1,   2,   4;
  1,   3,   9,  22;
  1,   4,  15,  52, 140;
etc.

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened.)
V. E. Hoggatt, Jr. and M. Bicknell, Catalan and related sequences arising from inverses of Pascal's triangle matrices, Fib. Quart., 14 (1976), 395-405.

Columns include A000012, A000027, A055999.
Right-hand diagonals include A002293, A069271, A006632.
Cf. A130458 (row sums).

A069270 := proc(n,k)
        binomial(n+3*k,k)*(n-k+1)/(n+2*k+1) ;
end proc: # R. J. Mathar, Oct 11 2015

Table[Binomial[n + 3 k, k] (n - k + 1)/(n + 2 k + 1), {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Dec 27 2019 *)

T(n, k) = C(n+3k, k)*(n-k+1)/(n+2k+1).
For n >= k+3: T(n, k) = T(n-2, k+1)-T(n-3, k+1).
T(n, n) = T(n+2, n-1) = C(4n, n)/(3n+1).

A130457 Triangle, read by rows of 3n+1 terms, where row n+1 is generated by taking partial sums of row n and then appending 2 zeros followed by the final term in the partial sums of row n, for n>=0, with T(0,0)=1.

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 1, 1, 2, 0, 0, 2, 1, 2, 3, 5, 5, 5, 7, 0, 0, 7, 1, 3, 6, 11, 16, 21, 28, 28, 28, 35, 0, 0, 35, 1, 4, 10, 21, 37, 58, 86, 114, 142, 177, 177, 177, 212, 0, 0, 212, 1, 5, 15, 36, 73, 131, 217, 331, 473, 650, 827, 1004, 1216, 1216, 1216, 1428, 0, 0, 1428, 1, 6, 21, 57

Offset: 0

Paul D. Hanna, May 26 2007

			Triangle begins:
.1;
.1, 0, 0, 1;
.1, 1, 1, 2, 0, 0, 2;
.1, 2, 3, 5, 5, 5, 7, 0, 0, 7;
.1, 3, 6, 11, 16, 21, 28, 28, 28, 35, 0, 0, 35;
.1, 4, 10, 21, 37, 58, 86, 114, 142, 177, 177, 177, 212, 0, 0, 212;
.1, 5, 15, 36, 73, 131, 217, 331, 473, 650, 827, 1004, 1216, 1216, 1216, 1428, 0, 0, 1428; ...

Cf. A130458 (final term in rows); A130456 (variant).

{T(n,k)=local(A=[1],B);if(n==0,if(k==0,1,0),for(j=1,n, B=Vec(Ser(A)/(1-x));A=concat(concat(B,[0,0]),B[ #B]));A[k+1])}

A376447 a(n) = Sum_{k=1..n} binomial(n(m+1)-mk-1, n-k)k/(nm-(m-1)*k), for m=7.

Original entry on oeis.org

1, 1, 2, 11, 113, 1472, 21375, 330905, 5348380, 89188027, 1522999490, 26497322726, 468008534344, 8369663126451, 151250089239576, 2757670822707609, 50665739960395844, 937090326420476362, 17433799590092921578, 326028128575350241133, 6125291348509146395197

Offset: 0

Vaclav Kotesovec, Sep 23 2024, following a suggestion from Mikhail Kurkov

nonn

Vaclav Kotesovec, Recurrence of order 8.

Cf. A000108 (m=1), A098746 (m=2), A130458 (m=3), A186182 (m=4), A186183 (m=5), A186184 (m=6).

Join[{1}, With[{m=7}, Table[Sum[Binomial[n*(m+1)-m*k-1, n-k]*k/(n*m-(m-1)*k), {k, 1, n}], {n, 1, 20}]]]

cp's OEIS Frontend

A069270 Third level generalization of Catalan triangle (0th level is Pascal's triangle A007318; first level is Catalan triangle A009766; 2nd level is A069269).

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A130457 Triangle, read by rows of 3n+1 terms, where row n+1 is generated by taking partial sums of row n and then appending 2 zeros followed by the final term in the partial sums of row n, for n>=0, with T(0,0)=1.

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PARI

A376447 a(n) = Sum_{k=1..n} binomial(n(m+1)-mk-1, n-k)k/(nm-(m-1)*k), for m=7.

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cp's OEIS Frontend

A069270 Third level generalization of Catalan triangle (0th level is Pascal's triangle A007318; first level is Catalan triangle A009766; 2nd level is A069269).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A130457 Triangle, read by rows of 3n+1 terms, where row n+1 is generated by taking partial sums of row n and then appending 2 zeros followed by the final term in the partial sums of row n, for n>=0, with T(0,0)=1.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

A376447 a(n) = Sum_{k=1..n} binomial(n*(m+1)-m*k-1, n-k)*k/(n*m-(m-1)*k), for m=7.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A376447 a(n) = Sum_{k=1..n} binomial(n(m+1)-mk-1, n-k)k/(nm-(m-1)*k), for m=7.