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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). 6
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 (list; table; graph; refs; listen; history; text; internal format)
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

REFERENCES

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

LINKS

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

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

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Last modified December 1 16:44 EST 2021. Contains 349430 sequences. (Running on oeis4.)