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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). 7
 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: A054450 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 October 13 19:35 EDT 2019. Contains 327981 sequences. (Running on oeis4.)