A124392 A Fine number related number triangle.

1, 2, 1, 7, 2, 1, 24, 8, 2, 1, 86, 28, 9, 2, 1, 314, 103, 32, 10, 2, 1, 1163, 382, 121, 36, 11, 2, 1, 4352, 1432, 456, 140, 40, 12, 2, 1, 16414, 5408, 1732, 536, 160, 44, 13, 2, 1, 62292, 20546, 6608, 2064, 622, 181, 48, 14, 2, 1, 237590, 78436, 25314, 7960, 2429, 714, 203, 52, 15, 2, 1

Offset: 0

Author

Paul Barry, Oct 30 2006

Keywords

• easy
• nonn
• tabl

Comments

First column is A014300. Second column is A114590. Row sums are A001700. Array is given by (f(x)/(x*sqrt(1-4x)), f(x)) where f(x) is g.f. of Fine numbers A000957.

Examples

			Triangle begins
      1;
      2,    1;
      7,    2,    1;
     24,    8,    2,   1;
     86,   28,    9,   2,   1;
    314,  103,   32,  10,   2,  1;
   1163,  382,  121,  36,  11,  2,  1;
   4352, 1432,  456, 140,  40, 12,  2, 1;
  16414, 5408, 1732, 536, 160, 44, 13, 2, 1;
		

Links

• G. C. Greubel, Rows n = 0..100 of triangle, flattened

Crossrefs

Cf. A000957, A001700, A014300, A114590.

Programs

GAP

Flat(List([0..10], n-> List([0..n], k-> Binomial(n-j, k)*Binomial(2*j, n-k) ))); # G. C. Greubel, Dec 25 2019

Magma

[(&+[Binomial(n-j, k)*Binomial(2*j, n-k): j in [0..n-k]]): k in [0..n], n in [0.10]]; // G. C. Greubel, Dec 25 2019

Maple

seq(seq( add(binomial(n-j, k)*binomial(2*j, n-k), j=0..n-k), k=0..n), n=0..10); # G. C. Greubel, Dec 25 2019

Mathematica

Table[Sum[Binomial[n-j, k]*Binomial[2*j, n-k], {j,0,n-k}], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 25 2019 *)

PARI

T(n,k) = sum(j=0, n-k, binomial(n-j, k)*binomial(2*j, n-k)); \\ G. C. Greubel, Dec 25 2019

Sage

[[sum(binomial(n-j, k)*binomial(2*j, n-k) for j in (0..n-k)) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Dec 25 2019

Formula

Riordan array ( 1/(x*sqrt(1-4*x)) * (1-sqrt(1-4*x))/(3-sqrt(1-4*x)), (1-sqrt(1-4*x))/(3-sqrt(1-4*x)) ).

Number triangle T(n, k) = Sum_{j=0..n-k} C(n-j, k)*C(2*j, n-k).