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A186332 Riordan array (1,x+x^2+x^3+x^4) without 0-column. 1
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, 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, 40, 135, 216, 203, 120, 45, 10, 1, 0, 0, 1, 31, 155, 336, 413, 322, 165, 55, 11, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

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

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]

LINKS

Table of n, a(n) for n=1..78.

Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.

FORMULA

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.

T(n,k)=sum(m=0..(n-k)/4, (-1)^m*binomial(k,k-m)*binomial(n-4*m-1,k-1)), n>0, n>=k.

EXAMPLE

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,

0,1,12,31,35,21,7,1,

0,0,10,40,65,56,28,8,1,

0,0,6,44,101,120,84,36,9,1,

0,0,3,40,135,216,203,120,45,10,1,

0,0,1,31,155,336,413,322,165,55,11,1.

MATHEMATICA

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

Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-Fran├žois Alcover, Jul 22 2018, from 2nd formula *)

CROSSREFS

Sequence in context: A048805 A204015 A216210 * A129571 A180180 A034931

Adjacent sequences:  A186329 A186330 A186331 * A186333 A186334 A186335

KEYWORD

nonn,easy,tabl

AUTHOR

Vladimir Kruchinin, Feb 17 2011

STATUS

approved

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Last modified October 16 01:33 EDT 2019. Contains 328038 sequences. (Running on oeis4.)