A356601 Triangle read by rows. T(n, k) = denominator(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, 2, 1, 6, 3, 1, 12, 3, 4, 1, 20, 30, 20, 5, 1, 30, 30, 10, 15, 6, 1, 42, 35, 70, 105, 14, 7, 1, 56, 7, 280, 35, 56, 7, 8, 1, 72, 252, 56, 630, 504, 28, 72, 9, 1, 90, 180, 105, 630, 126, 420, 45, 45, 10, 1, 110, 495, 33, 1155, 1386, 1155, 165, 99, 110, 11, 1

Offset

0,2

Links

Table of n, a(n) for n=0..65.

Grzegorz Rządkowski, Bernoulli numbers and solitons - revisited, Journal of Nonlinear Mathematical Physics, (2010) 17:1, 121-126.

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) = denominator(R(n, k)).

Example

Triangle T(n, k) starts:
[0]  1;
[1]  2,   1;
[2]  6,   3,   1;
[3] 12,   3,   4,   1;
[4] 20,  30,  20,   5,   1;
[5] 30,  30,  10,  15,   6,   1;
[6] 42,  35,  70, 105,  14,   7,  1;
[7] 56,   7, 280,  35,  56,   7,  8,  1;
[8] 72, 252,  56, 630, 504,  28, 72,  9,  1;
[9] 90, 180, 105, 630, 126, 420, 45, 45, 10,  1;
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(denom(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[Denominator[T[n, k]], {n, 0, 8}, {k, 0, n}] // TableForm

Crossrefs

Cf. A356602 (numerator), A173018, A278075, A356545, A356547.

Sequence in context: A065553 A016545 A142977 * A362997 A120108 A350292

Adjacent sequences: A356598 A356599 A356600 * A356602 A356603 A356604

Keyword

nonn,tabl,frac

Author

Peter Luschny, Aug 15 2022

Status

approved