A109970 -id:A109970 - cp's OEIS frontend

A109971 Inverse of Riordan array (1,x(1-x)^2), A109970.

1, 0, 1, 0, 2, 1, 0, 7, 4, 1, 0, 30, 18, 6, 1, 0, 143, 88, 33, 8, 1, 0, 728, 455, 182, 52, 10, 1, 0, 3876, 2448, 1020, 320, 75, 12, 1, 0, 21318, 13566, 5814, 1938, 510, 102, 14, 1, 0, 120175, 76912, 33649, 11704, 3325, 760, 133, 16, 1, 0, 690690, 444015, 197340

Offset: 0

Paul Barry, Jul 06 2005

Row sums are A001764. Diagonal sums are A109972. Second column is A006013. Third column is A006629.

			Rows begin
1;
0,1;
0,2,1;
0,7,4,1;
0,30,18,6,1;
0,143,88,33,8,1;
Production array begins
0, 1
0, 2, 1
0, 3, 2, 1
0, 4, 3, 2, 1
0, 5, 4, 3, 2, 1
0, 6, 5, 4, 3, 2, 1,
0, 7, 6, 5, 4, 3, 2, 1
0, 8, 7, 6, 5, 4, 3, 2, 1
0, 9, 8, 7, 6, 5, 4, 3, 2, 1
... - _Philippe Deléham_, Mar 05 2013

Naiomi Cameron, J. E. McLeod, Returns and Hills on Generalized Dyck Paths, Journal of Integer Sequences, Vol. 19, 2016, #16.6.1.
W.-j. Woan, The Lagrange inversion formula and divisibility properties, JIS 10 (2007) 07.7.8, example 4.

Essentially the same as A092276.

Number triangle T(0, 0)=1, T(0, k)=0, k>0, T(n, k)=(k/n)*binomial(3n-k-1, n-k) otherwise; Riordan array (1, f) where f(1-f)^2=x.
T(n, k)=sum{j=0..n, ((3j+1)/(2n+j+1))(-1)^(j-k)*C(3n, 2n+j)C(j, k)}; - Paul Barry, Oct 07 2005
T(n,k)=binomial(3n-k,n-k)*2k/(3n-k). (Paul Barry, May 18 2006)

A109972 Diagonal sums of number triangle A109970.

Original entry on oeis.org

1, 0, 1, 2, 8, 34, 162, 822, 4365, 23956, 134814, 773746, 4511693, 26652346, 159170385, 959412290, 5829083420, 35661048886, 219491344362, 1358204062536, 8444658457530, 52729475008690, 330518562341537, 2078987880103170

Offset: 0

Paul Barry, Jul 06 2005

Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.

a(n) = if (n==0, 1, sum(k=0, n-1, (k/(n-k))*binomial(3*n-4*k-1, n-2*k))); \\ Michel Marcus, Jan 25 2019

a(0) = 1, a(n) = Sum_{k=0..n-1} (k/(n-k))*binomial(3n-4k-1, n-2k), n>0. [corrected by Michel Marcus, Jan 25 2019]

A116088 Riordan array (1, x*(1+x)^2).

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 1, 4, 1, 0, 0, 6, 6, 1, 0, 0, 4, 15, 8, 1, 0, 0, 1, 20, 28, 10, 1, 0, 0, 0, 15, 56, 45, 12, 1, 0, 0, 0, 6, 70, 120, 66, 14, 1, 0, 0, 0, 1, 56, 210, 220, 91, 16, 1, 0, 0, 0, 0, 28, 252, 495, 364, 120, 18, 1

Offset: 0

Paul Barry, Feb 04 2006

			Triangle begins as:
  1;
  0, 1;
  0, 2, 1;
  0, 1, 4,  1;
  0, 0, 6,  6,  1;
  0, 0, 4, 15,  8,  1;
  0, 0, 1, 20, 28, 10,  1;
  0, 0, 0, 15, 56, 45, 12, 1;

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (Rows 0 <= n <= 150).
Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.

Row sums are A002478. Diagonal sums are A094686. Inverse is (-1)^(n-k) * A109971(n,k). Unsigned version of A109970.

Flat(List([0..10], n->List([0..n], k-> Binomial(2*k, n-k) ))); # G. C. Greubel, May 09 2019

[[Binomial(2*k, n-k): k in [0..n]]: n in [0..10]]; // G. C. Greubel, May 09 2019

Flatten[Table[Binomial[2k,n-k],{n,0,20},{k,0,n}]] (* Harvey P. Dale, Oct 22 2012 *)

{T(n,k) = binomial(2*k, n-k)}; \\ G. C. Greubel, May 09 2019

[[binomial(2*k, n-k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, May 09 2019

G.f.: 1/(1-x*y*(1+x)^2).

Number triangle T(n,k) = C(2*k, n-k) = C(n,k)*C(3*k,n)/C(3*k,k).

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A109971 Inverse of Riordan array (1,x(1-x)^2), A109970.

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A109972 Diagonal sums of number triangle A109970.

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A116088 Riordan array (1, x*(1+x)^2).

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