A053538 Triangle: a(n,m) = ways to place p balls in n slots with m in the rightmost p slots, 0<=p<=n, 0<=m<=n, summed over p, a(n,m)= Sum_{k=0..n} binomial(k,m)*binomial(n-k,k-m), (see program line).

1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 5, 4, 1, 1, 8, 10, 7, 5, 1, 1, 13, 18, 16, 9, 6, 1, 1, 21, 33, 31, 23, 11, 7, 1, 1, 34, 59, 62, 47, 31, 13, 8, 1, 1, 55, 105, 119, 101, 66, 40, 15, 9, 1, 1, 89, 185, 227, 205, 151, 88, 50, 17, 10, 1, 1, 144, 324, 426, 414, 321, 213, 113, 61, 19, 11, 1, 1

Offset

0,4

Comments

Riordan array (1/(1-x-x^2), x(1-x)/(1-x-x^2)). Row sums are A000079. Diagonal sums are A006053(n+2). - Paul Barry, Nov 01 2006

Subtriangle of the triangle given by (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 05 2012

Mirror image of triangle in A208342. - Philippe Deléham, Mar 05 2012

A053538 is jointly generated with A076791 as an array of coefficients of polynomials u(n,x): initially, u(1,x)=v(1,x)=1, for n>1, u(n,x) = x*u(n-1,x) + v(n-1,x) and v(n,x) = u(n-1,x) + v(n-1,x). See the Mathematica section at A076791. - Clark Kimberling, Mar 08 2012

The matrix inverse starts

1;

-1, 1;

-1, -1, 1;

1, -2, -1, 1;

3, 1, -3, -1, 1;

1, 6, 1, -4, -1, 1;

-7, 4, 10, 1, -5, -1, 1;

-13, -13, 8, 15, 1, -6, -1, 1;

3, -31, -23, 13, 21, 1, -7, -1, 1; - R. J. Mathar, Mar 15 2013

Links

Alois P. Heinz, Rows n = 0..140, flattened

R. P. Grimaldi, Extraordinary subsets: a generalization, Fib. Quart., 55 (No. 3, 2017), 114-122. See Table 1.

Formula

From Philippe Deléham, Mar 05 2012: (Start)

T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) - T(n-2,k-1), T(0,0) = T(1,0) = T(1,1) = 1 and T(n,k) = 0 if k<0 or if k>n.

G.f.: 1/(1-(1+y)*x-(1-y)*x^2).

Sum_{k, 0<=k<=n} T(n,k)*x^k = A077957(n), A000045(n+1), A000079(n), A001906(n+1), A007070(n), A116415(n), A084326(n+1), A190974(n+1), A190978(n+1), A190984(n+1), A190990(n+1), A190872(n+1) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 respectively. (End)

Example

n=4; Table[binomial[k, j]binomial[n-k, k-j], {k, 0, n}, {j, 0, n}] splits {1, 4, 6, 4, 1} into {{1, 0, 0, 0, 0}, {3, 1, 0, 0, 0}, {1, 4, 1, 0, 0}, {0, 0, 3, 1, 0}, {0, 0, 0, 0, 1}} and this gives summed by columns {5, 5, 4, 1, 1}
Triangle begins :
   1;
   1,  1;
   2,  1,  1;
   3,  3,  1, 1;
   5,  5,  4, 1, 1;
   8, 10,  7, 5, 1, 1;
  13, 18, 16, 9, 6, 1, 1;
...
(0, 1, 1, -1, 0, 0, 0, ...) DELTA (1, 0, -1, 1, 0, 0, 0, ...) begins :
  1;
  0,  1;
  0,  1,  1;
  0,  2,  1,  1;
  0,  3,  3,  1, 1;
  0,  5,  5,  4, 1, 1;
  0,  8, 10,  7, 5, 1, 1;
  0, 13, 18, 16, 9, 6, 1, 1;

Maple

a:= (n, m)-> add(binomial(k, m)*binomial(n-k, k-m), k=0..n):
seq(seq(a(n, m), m=0..n), n=0..12);  # Alois P. Heinz, Sep 19 2013

Mathematica

Table[Sum[Binomial[k, m]*Binomial[n-k, k-m], {k, 0, n}], {n, 0, 12}, {m, 0, n}]

Prog

(PARI) {T(n, k) = sum(j=0, n, binomial(j, k)*binomial(n-j, j-k))}; \\ G. C. Greubel, May 16 2019
(Magma) [[(&+[Binomial(j, k)*Binomial(n-j, j-k): j in [0..n]]): k in [0..n]]: n in [0..12]]; // G. C. Greubel, May 16 2019
(Sage) [[sum(binomial(j, k)*binomial(n-j, j-k) for j in (0..n)) for k in (0..n)] for n in (0..12)] # G. C. Greubel, May 16 2019
(GAP) Flat(List([0..12], n-> List([0..n], k-> Sum([0..n], j->  Binomial(j, k)*Binomial(n-j, j-k)) ))); # G. C. Greubel, May 16 2019

Crossrefs

Cf. A000045, A000079, A208342.

Sequence in context: A337009 A174802 A238346 * A235803 A138201 A220614

Adjacent sequences: A053535 A053536 A053537 * A053539 A053540 A053541

Keyword

nonn,tabl

Author

Wouter Meeussen, May 23 2001

Status

approved