A342313 -id:A342313 - cp's OEIS frontend

A342312 T(n, k) = ((2n + 1)/2)Sum_{j, k, n} (-1)^(k + j)(n + j)binomial(2n, n - j) Stirling2(n - k + j, 1 - k + j) with T(0, 0) = 1. Triangle read by row, T(n, k) for 0 <= k <= n.

1, 0, 3, 0, 0, 10, 0, 0, -42, 21, 0, 0, 216, -288, 36, 0, 0, -1320, 3190, -1210, 55, 0, 0, 9360, -34632, 25584, -4056, 78, 0, 0, -75600, 389340, -462000, 152460, -11970, 105, 0, 0, 685440, -4621824, 7907040, -4368320, 762960, -32640, 136

Offset: 0

Peter Luschny, Mar 08 2021

The triangle can be seen as representing the numerators of a sequence of rational polynomials. Let p_{n}(x) = Sum_{k=0..n} (T(n, k)/A342313(n, k))*x^k. Then p_{n}(1) = B_{n}(1), where B_{n}(x) are the Bernoulli polynomials.

			The triangle starts:
  [0] 1
  [1] 0, 3
  [2] 0, 0,    10
  [3] 0, 0,   -42,     21
  [4] 0, 0,   216,   -288,    36
  [5] 0, 0, -1320,   3190, -1210,    55
  [6] 0, 0,  9360, -34632, 25584, -4056, 78
The first few polynomials are P(n, k) = T(n, k) / A342313(n, k):
  1;
  0, 1/2;
  0,  0,  1/6;
  0,  0, -1/10,   1/10;
  0,  0,  3/35,  -4/21,   1/14;
  0,  0, -2/21,  29/84, -11/36,    1/18;
  0,  0, 10/77, -37/55, 164/165, -26/55, 1/22;

Levin Luschny, Illustrating the polynomials.

Cf. A014105 (main diagonal), A342313 (denominators), A340556.

T := (n, k) -> `if`(n = 0 and k = 0, 1, (n+1/2)*
add((-1)^j*(n+k+j)*binomial(2*n, n-k-j)*Stirling2(n + j, j + 1) , j= 0..n-k)):
seq(print(seq(T(n, k), k=0..n)), n=0..6);

T[0, 0] := 1; T[n_, k_] := ((2n + 1)/2) Sum[(-1)^(k+j)(n+j) Binomial[2n, n-j] StirlingS2[n-k+j, 1-k+j], {j, k, n}];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten

T(n, k) = numerator([x^k] p(n, x)) for n >= 2, where p(n, x) = (1/2)*Sum_{k=0..n-1} (-1)^k*x^(n-k)*E2(n - 1, k + 1) / binomial(2*n - 1, k + 1) and E2(n,k) denotes the second-order Eulerian numbers A340556.

Another representation of the polynomials for n >= 2 is p(n, x) = (1/2)*Sum_{k=0..n} x^k*Sum_{j=k..n} ((-1)^(j + k)*((n - k + 1)!*(n + k - 2)!)/((j + n - 1)!*(n - j)!))*Stirling2(n - k + j, j - k + 1).

A342321 T(n, k) = A343277(n)[x^k] p(n, x) where p(n, x) = (1/(n+1))Sum_{k=0..n} (-1)^kE1(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

Peter Luschny, Mar 09 2021

Conjecture: For even n >= 6 p(n, x)/x and for odd n >= 3 p(n, x)/(x^2 - x) is irreducible.

			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.

Peter Luschny, Illustration of the polynomials.
Peter Luschny, Eulberian polynomials, A notebook companion to A342321 and A356601, 2022.

Cf. A343277, A123125.
Sequences of rational polynomials p(n, x) with p(n, 1) = Bernoulli(n, 1):
A342312/A342313, A342321/A342320, A342322/A064538.

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);

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

An alternative representation of the sequence of rational polynomials is:
p(n, x) = Sum_{k=1..n} x^k*k!*Sum_{j=0..k} (-1)^(n-j)*Stirling2(n, j)/((k - j)!(n - j + 1)*binomial(n + 1, j)) for n >= 1 and p(0, x) = 1.
(Sum_{k = 0..n} T(n, k)) / A343277(n) = Bernoulli(n, 1).

cp's OEIS Frontend

A342312 T(n, k) = ((2n + 1)/2)Sum_{j, k, n} (-1)^(k + j)(n + j)binomial(2n, n - j) Stirling2(n - k + j, 1 - k + j) with T(0, 0) = 1. Triangle read by row, T(n, k) for 0 <= k <= n.

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A342321 T(n, k) = A343277(n)[x^k] p(n, x) where p(n, x) = (1/(n+1))Sum_{k=0..n} (-1)^kE1(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.

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cp's OEIS Frontend

A342312 T(n, k) = ((2*n + 1)/2)*Sum_{j, k, n} (-1)^(k + j)*(n + j)*binomial(2*n, n - j)* Stirling2(n - k + j, 1 - k + j) with T(0, 0) = 1. Triangle read by row, T(n, k) for 0 <= k <= n.

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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.

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Formula

A342312 T(n, k) = ((2n + 1)/2)Sum_{j, k, n} (-1)^(k + j)(n + j)binomial(2n, n - j) Stirling2(n - k + j, 1 - k + j) with T(0, 0) = 1. Triangle read by row, T(n, k) for 0 <= k <= n.

A342321 T(n, k) = A343277(n)[x^k] p(n, x) where p(n, x) = (1/(n+1))Sum_{k=0..n} (-1)^kE1(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.