A356602 Triangle read by rows. T(n, k) = numerator(Integral_{z=0..1} Eulerian(n, k)*z^(k + 1)*(z - 1)^(n - k - 1) dz), where Eulerian(n, k) = A173018(n, k) for n >= 1, and T(0, 0) = 1.

1, 1, 0, -1, 1, 0, 1, -1, 1, 0, -1, 11, -11, 1, 0, 1, -13, 11, -13, 1, 0, -1, 19, -151, 302, -19, 1, 0, 1, -5, 1191, -302, 397, -15, 1, 0, -1, 247, -477, 15619, -15619, 477, -247, 1, 0, 1, -251, 1826, -44117, 15619, -44117, 1826, -251, 1, 0

Offset

0,12

Links

Table of n, a(n) for n=0..54.

Formula

R(n, k) = (-1)^(k - n + 1)*Eulerian(n, k)*Gamma(k + 2)*Gamma(n - k)/Gamma(n + 2) for 0 <= k < n, and T(n, n) = 0^n.

Bernoulli(n) = Sum_{k=0..n} R(n, k), where Bernoulli(1) = 1/2.

T(n, k) = numerator(R(n, k)).

Example

Triangle T(n, k) starts:
[0]  1;
[1]  1,    0;
[2] -1,    1,    0;
[3]  1,   -1,    1,      0;
[4] -1,   11,  -11,      1,      0;
[5]  1,  -13,   11,    -13,      1,      0;
[6] -1,   19, -151,    302,    -19,      1,    0;
[7]  1,   -5, 1191,   -302,    397,    -15,    1,    0;
[8] -1,  247, -477,  15619, -15619,    477, -247,    1,  0;
[9]  1, -251, 1826, -44117,  15619, -44117, 1826, -251,  1,  0;
The Bernoulli numbers (with B(1) = 1/2) are the row sums of the fractions.
[0]   1                                              =     1;
[1] + 1/2                                            =   1/2;
[2] - 1/6  +  1/3                                    =   1/6;
[3] + 1/12 -  1/3  +    1/4                          =     0;
[4] - 1/20 + 11/30 -  11/20 +   1/5                  = -1/30;
[5] + 1/30 - 13/30 +  11/10 -  13/15  +   1/6        =     0;
[6] - 1/42 + 19/35 - 151/70 + 302/105 - 19/14 + 1/7  =  1/42;

Maple

E1 := proc(n, k) combinat:-eulerian1(n, k) end:
Trow := proc(n, z) if n = 0 then return 1 fi;
seq(numer(int(E1(n, k)*z^(k + 1)*(z - 1)^(n - k - 1), z=0..1)), k=0..n) end:
for n from 0 to 9 do Trow(n, z) od;

Mathematica

Unprotect[Power]; Power[0, 0] = 1;
E1[n_, k_] /; n == k = 0^k; E1[n_, k_] /; k < 0 || k > n = 0;
E1[n_, k_] := E1[n, k] = (k + 1)*E1[n - 1, k] + (n - k)*E1[n - 1, k - 1];
T[n_, k_] /; n == k = 0^k;
T[n_, k_] := (-1)^(k - n + 1)*E1[n, k]*Gamma[k + 2]*Gamma[n - k]/Gamma[n + 2];
Table[Numerator[T[n, k]], {n, 0, 8}, {k, 0, n}] // TableForm

Crossrefs

Cf. A356601 (denominator), A173018, A278075, A356545, A356547.

Sequence in context: A167262 A288393 A288642 * A014459 A046609 A262014

Adjacent sequences: A356599 A356600 A356601 * A356603 A356604 A356605

Keyword

sign,tabl,frac

Author

Peter Luschny, Aug 15 2022

Status

approved