A342313
T(n, k) = (n + k - 1)*(n + k)*binomial(2*n + 1, n - k + 1) with T(0, 0) = T(1, 0) = 1. Triangle read by rows, T(n, k) for 0 <= k <= n.
Original entry on oeis.org
1, 1, 6, 20, 60, 60, 210, 420, 420, 210, 1512, 2520, 2520, 1512, 504, 9240, 13860, 13860, 9240, 3960, 990, 51480, 72072, 72072, 51480, 25740, 8580, 1716, 270270, 360360, 360360, 270270, 150150, 60060, 16380, 2730, 1361360, 1750320, 1750320, 1361360, 816816, 371280, 123760, 28560, 4080
Offset: 0
Triangle starts:
[0] 1
[1] 1, 6
[2] 20, 60, 60
[3] 210, 420, 420, 210
[4] 1512, 2520, 2520, 1512, 504
[5] 9240, 13860, 13860, 9240, 3960, 990
[6] 51480, 72072, 72072, 51480, 25740, 8580, 1716
[7] 270270, 360360, 360360, 270270, 150150, 60060, 16380, 2730
[8] 1361360, 1750320, 1750320, 1361360, 816816, 371280, 123760, 28560, 4080
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T := (n, k) -> `if`(n=0, 1,`if`(n=1 and k=0, 1,
(n + k - 1)*(n + k)*binomial(2*n + 1, n - k + 1))):
seq(print(seq(T(n, k), k = 0..n)), n = 0..8);
-
T[0, 0] := 1; T[1, 0] := 1;
T[n_, k_] := (n - 1 + k) (n + k) Binomial[2n + 1, n - k + 1];
Table[T[n, k], {n, 0, 8}, {k, 0, n}]
A342321
T(n, k) = A343277(n)*[x^k] p(n, x) where p(n, x) = (1/(n+1))*Sum_{k=0..n} (-1)^k*E1(n, k)*x^(n - k) / binomial(n, k), and E1(n, k) are the Eulerian numbers A123125. Triangle read by rows, for 0 <= k <= n.
Original entry on oeis.org
1, 0, 1, 0, -1, 2, 0, 1, -4, 3, 0, -3, 22, -33, 12, 0, 1, -13, 33, -26, 5, 0, -5, 114, -453, 604, -285, 30, 0, 5, -200, 1191, -2416, 1985, -600, 35, 0, -35, 2470, -21465, 62476, -78095, 42930, -8645, 280, 0, 14, -1757, 21912, -88234, 156190, -132351, 51128, -7028, 126
Offset: 0
Triangle starts:
[n] T(n, k) A343277(n)
----------------------------------------------------------
[0] 1; [1]
[1] 0, 1; [2]
[2] 0, -1, 2; [6]
[3] 0, 1, -4, 3; [12]
[4] 0, -3, 22, -33, 12; [60]
[5] 0, 1, -13, 33, -26, 5; [30]
[6] 0, -5, 114, -453, 604, -285, 30; [210]
[7] 0, 5, -200, 1191, -2416, 1985, -600, 35; [280]
.
The coefficients of the polynomials p(n, x) = (Sum_{k = 0..n} T(n, k) x^k) / A343277(n) for the first few n:
[0] 1;
[1] 0, 1/2;
[2] 0, -1/6, 1/3;
[3] 0, 1/12, -1/3, 1/4;
[4] 0, -1/20, 11/30, -11/20, 1/5;
[5] 0, 1/30, -13/30, 11/10, -13/15, 1/6.
Sequences of rational polynomials p(n, x) with p(n, 1) = Bernoulli(n, 1):
-
CoeffList := p -> op(PolynomialTools:-CoefficientList(p,x)):
E1 := (n, k) -> combinat:-eulerian1(n, k):
poly := n -> (1/(n+1))*add((-1)^k*E1(n,k)*x^(n-k)/binomial(n,k), k=0..n):
Trow := n -> denom(poly(n))*CoeffList(poly(n)): seq(Trow(n), n = 0..9);
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Poly342321[n_, x_] := If[n == 0, 1, Sum[x^k*k!*Sum[(-1)^(n - j)*StirlingS2[n, j] /((k - j)!(n - j + 1) Binomial[n + 1, j]), {j, 0, k}], {k, 1, n}]];
Table[A343277[n] CoefficientList[Poly342321[n, x], x][[k+1]], {n, 0, 9}, {k, 0, n}] // Flatten
Showing 1-2 of 2 results.
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