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

0,2

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.

Links

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

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).

Example

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;

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 *)

Prog

(PARI) T(n, k) = sum(j=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
(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
(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

Crossrefs

Cf. A000957, A001700, A014300, A114590.

Sequence in context: A341738 A128747 A205945 * A144446 A121416 A317547

Adjacent sequences: A124389 A124390 A124391 * A124393 A124394 A124395

Keyword

easy,nonn,tabl

Author

Paul Barry, Oct 30 2006

Status

approved