A071922 Unimodal analog of binomial coefficient, such that A071921(n,m) = a(n+m-1,n) for all (n,m) different from (0,0), arranged in a Pascal-like triangle.

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 9, 7, 1, 1, 5, 16, 22, 11, 1, 1, 6, 25, 50, 46, 16, 1, 1, 7, 36, 95, 130, 86, 22, 1, 1, 8, 49, 161, 295, 296, 148, 29, 1, 1, 9, 64, 252, 581, 791, 610, 239, 37, 1, 1, 10, 81, 372, 1036, 1792, 1897, 1163, 367, 46, 1, 1, 11, 100, 525, 1716, 3612

Offset: 0

Author

Michele Dondi (bik.mido(AT)tiscalinet.it), Jun 14 2002

Keywords

• nonn
• easy
• tabl

Comments

Also, number of n-length k-ary words avoiding the pattern 1'-2-1". - Ralf Stephan, Apr 28 2004

The matrix inverse starts

1;

-1, 1;

1, -2, 1;

-2, 5, -4, 1;

8, -21, 19, -7, 1;

-56, 148, -137, 55, -11, 1;

608, -1608, 1493, -608, 130, -16, 1;

-9440, 24968, -23190, 9461, -2044, 266, -22, 1;

198272, -524416, 487088, -198761, 42997, -5642, 490, -29, 1; - R. J. Mathar, Mar 15 2013

Examples

			Triangle begins
  1;
  1,   1;
  1,   2,   1;
  1,   3,   4,   1;
  1,   4,   9,   7,   1;
  1,   5,  16,  22,  11,   1;
  1,   6,  25,  50,  46,  16,   1;
  1,   7,  36,  95, 130,  86,  22,   1;
		

Links

• G. C. Greubel, Rows n = 0..100 of triangle, flattened
• S. Kitaev and T. Mansour, Partially ordered generalized patterns and k-ary words, arXiv:math/0210023 [math.CO], 2003.
• S. Kitaev and T. Mansour, Partially ordered generalized patterns and k-ary words, Annals of Combinatorics, 7(2) (2003), 191-200.

Programs

GAP

Flat(List([0..11], n-> List([0..n], k-> Sum([0..n-k], j-> Binomial( 2*j+k-1, 2*j) )))); # G. C. Greubel, Aug 26 2019

Magma

[&+[Binomial(2*j+k-1, 2*j): j in [0..n-k]]: k in [0..n], n in [0..11]]; // G. C. Greubel, Aug 26 2019

Maple

A071922 := proc(n,k)
    add( binomial(2*j+k-1,2*j),j=0..n-k) ;
end proc: # R. J. Mathar, Mar 15 2013

Mathematica

a[n_, m_]:= Sum[Binomial[2k+m-1, 2k], {k, 0, n-m}]; Flatten[ Table[ a[n, m], {n, 0, 11}, {m, 0, n}]]

PARI

a(n,k) = sum(j=0, n-k, binomial(2*j+k-1, 2*j));
for(n=0,11, for(k=0,n, print1(a(n,k), ", "))) \\ G. C. Greubel, Aug 26 2019

Sage

[[sum(binomial(2*j+k-1, 2*j) for j in (0..n-k)) for k in (0..n)] for n in (0..11)] # G. C. Greubel, Aug 26 2019

Formula

a(n, m) = Sum_{k=0..n-m} binomial(2*k+m-1, 2*k).

Sum_{m=0..n} a(n, m) = 1 + Fibonacci(2*n).

Sum_{m=0..n} (-1)^m*a(n, m) = 1 if 3 divides n, 0 otherwise.

G.f. for k-th row: 1/(1-x)^(2k-1) + Sum_{j=1..k-1} x/(1-x)^(2j). - Ralf Stephan, Apr 28 2004

Extensions

Edited by Robert G. Wilson v, Jun 17 2002