A186332 - OEIS

%I #27 Feb 04 2021 03:23:09

%S 1,1,1,1,2,1,1,3,3,1,0,4,6,4,1,0,3,10,10,5,1,0,2,12,20,15,6,1,0,1,12,

%T 31,35,21,7,1,0,0,10,40,65,56,28,8,1,0,0,6,44,101,120,84,36,9,1,0,0,3,

%U 40,135,216,203,120,45,10,1,0,0,1,31,155,336,413,322,165,55,11,1

%N Riordan array (1, x + x^2 + x^3 + x^4) without 0-column.

%C Columns k >= 1 contain the expansion coefficients T(n,k) = [x^(n-k)] (x + x^2 + x^3 + x^4)^k.

%C Number of lattice paths from (0,0) to (n,k) using steps (1,1), (2,1), (3,1), (4,1). - _Joerg Arndt_, Jul 05 2011

%H Vladimir Kruchinin, <a href="http://arxiv.org/abs/1009.2565">Composition of ordinary generating functions</a>, arXiv:1009.2565 [math.CO], 2010.

%F T(n,k) = Sum_{j=0..k} binomial(k,j) * Sum_{i=0..n-k} binomial(j,i)*binomial(k-j,n-3*k+2*j-i), n>0, n>=k.

%F T(n,k) = Sum_{m=0..floor((n-k)/4)} (-1)^m*binomial(k,k-m)*binomial(n-4*m-1,k-1), n>0, n>=k.

%F O.g.f. of row polynomials R(n, x). I.e., o.g.f. of triangle (Riordan): G(z,x) = 1/(1 - x*z*(1+z)*(1+z^2)) - 1 (without column k=0). - _Wolfdieter Lang_, Jan 29 2021

%e Array begins

%e   1;

%e   1, 1;

%e   1, 2,  1;

%e   1, 3,  3,  1;

%e   0, 4,  6,  4,   1;

%e   0, 3, 10, 10,   5,   1;

%e   0, 2, 12, 20,  15,   6,   1;

%e   0, 1, 12, 31,  35,  21,   7,   1;

%e   0, 0, 10, 40,  65,  56,  28,   8,   1;

%e   0, 0,  6, 44, 101, 120,  84,  36,   9,  1;

%e   0, 0,  3, 40, 135, 216, 203, 120,  45, 10,  1;

%e   0, 0,  1, 31, 155, 336, 413, 322, 165, 55, 11, 1;

%e   ...

%t T[n_, k_] := Sum[(-1)^m*Binomial[k, k - m]*Binomial[n - 4*m - 1, k - 1], {m, 0, (n - k)/4}];

%t Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jul 22 2018, from 2nd formula *)

%K nonn,easy,tabl

%O 1,5

%A _Vladimir Kruchinin_, Feb 17 2011