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

0,5

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

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.

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

Example

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;

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}]]

Prog

(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
(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
(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
(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

Crossrefs

Sequence in context: A122084 A104559 A080853 * A138028 A009999 A322268

Adjacent sequences: A071919 A071920 A071921 * A071923 A071924 A071925

Keyword

nonn,easy,tabl

Author

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

Extensions

Edited by Robert G. Wilson v, Jun 17 2002

Status

approved