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%I #14 Feb 15 2018 20:36:16
%S 1,1,1,2,2,1,3,7,3,1,5,16,15,4,1,8,38,46,26,5,1,13,82,141,100,40,6,1,
%T 21,173,381,375,185,57,7,1,34,352,983,1216,820,308,77,8,1,55,701,2400,
%U 3704,3101,1575,476,100,9,1,89,1368,5646,10536,10885,6804,2758,696,126,10,1
%N Triangle read by rows, T(n,k) = [x^k] Sum_{k=0..n} p_{n,k}(x) where p_{n,k}(x) = x^(n-k)*binomial(n,k)*hypergeom([-k, k-n, k-n], [1, -n], 1/x), for 0 <= k <= n.
%F Let P_{n}(x) = Sum_{k=0..n} p_{n,k}(x) then
%F 2^n*P_{n}(1/2) = A299502(n).
%F P_{n}(-1) = A182883(n). P_{n}(0) = A000045(n+1).
%F P_{n}(1) = A108626(n). P_{n}(2) = A299501(n).
%F The general case: for fixed k the sequence P_{n}(k) with n >= 0 has the generating function ogf(k, x) = (1-2*(k+1)*x + (k^2+2*k-1)*x^2 - 2*(k-1)*x^3 + x^4)^(-1/2). The example section shows the start of this square array of sequences.
%F These sequences can be computed by the recurrence P(n,k) = ((2-n)*P(n-4,k)-(3-2*n)*(k-1)*P(n-3,k)+(k^2+2*k-1)*(1-n)*P(n-2,k)+(2*n-1)*(k+1)*P(n-1,k))/n with initial values 1, k+1, (k+1)^2+1 and (k+1)^3+4*k+2.
%F The partial polynomials p_{n,k}(x) reduce for x = 1 to A108625 (seen as a triangle).
%e The partial polynomials p_{n,k}(x) start:
%e [0] 1
%e [1] x, 1
%e [2] x^2, 2*x+1, 1
%e [3] x^3, 3*x^2+4*x, 3*x+2, 1
%e [4] x^4, 4*x^3+9*x^2, 6*x^2+12*x+1, 4*x+3, 1
%e [5] x^5, 5*x^4+16*x^3, 10*x^3+36*x^2+9*x, 10*x^2+24*x+3, 5*x+4, 1
%e .
%e The polynomials P_{n}(x) start:
%e [0] 1
%e [1] 1 + x
%e [2] 2 + 2*x + x^2
%e [3] 3 + 7*x + 3*x^2 + x^3
%e [4] 5 + 16*x + 15*x^2 + 4*x^3 + x^4
%e [5] 8 + 38*x + 46*x^2 + 26*x^3 + 5*x^4 + x^5
%e .
%e The triangle starts:
%e [0] 1
%e [1] 1, 1
%e [2] 2, 2, 1
%e [3] 3, 7, 3, 1
%e [4] 5, 16, 15, 4, 1
%e [5] 8, 38, 46, 26, 5, 1
%e [6] 13, 82, 141, 100, 40, 6, 1
%e [7] 21, 173, 381, 375, 185, 57, 7, 1
%e [8] 34, 352, 983, 1216, 820, 308, 77, 8, 1
%e [9] 55, 701, 2400, 3704, 3101, 1575, 476, 100, 9, 1'
%e .
%e The square array P_{n}(k) near k=0:
%e ...... [k=-2] 1, -1, 2, -7, 17, -44, 125, -345, 958, -2707, ...
%e A182883 [k=-1] 1, 0, 1, -2, 1, -6, 7, -12, 31, -40, ...
%e A000045 [k=0] 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...
%e A108626 [k=1] 1, 2, 5, 14, 41, 124, 383, 1200, 3799, 12122, ...
%e A299501 [k=2] 1, 3, 10, 37, 145, 588, 2437, 10251, 43582, 186785, ...
%e ...... [k=3] 1, 4, 17, 78, 377, 1886, 9655, 50220, 264223, 1402108, ...
%p CoeffList := p -> op(PolynomialTools:-CoefficientList(p,x)):
%p PrintPoly := p -> print(sort(expand(p),x,ascending)):
%p T := (n,k) -> x^(n-k)*binomial(n, k)*hypergeom([-k,k-n,k-n], [1,-n], 1/x):
%p P := [seq(add(simplify(T(n,k)),k=0..n), n=0..10)]:
%p seq(CoeffList(p), p in P); # seq(PrintPoly(p), p in P);
%p R := proc(n,k) option remember; # Recurrence
%p if n < 4 then return [1,k+1,(k+1)^2+1,(k+1)^3+4*k+2][n+1] fi; ((2-n)*R(n-4,k)-
%p (3-2*n)*(k-1)*R(n-3,k)+(k^2+2*k-1)*(1-n)*R(n-2,k)+(2*n-1)*(k+1)*R(n-1,k))/n end:
%p for k from -2 to 3 do lprint(seq(R(n,k), n=0..9)) od;
%Y Cf. A000045, A108625, A108626, A182883, A299499, A299501, A299502.
%K nonn,tabl
%O 0,4
%A _Peter Luschny_, Feb 11 2018
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