A356547 -id:A356547 - cp's OEIS frontend

A356545 Triangle read by rows. T(n, k) are the coefficients of polynomials p_n(x) based on the Eulerian numbers of first order representing the Bernoulli numbers as B_n = p_n(1) / (n + 1)!.

1, 1, 0, 2, -1, 0, 6, -8, 2, 0, 24, -66, 44, -6, 0, 120, -624, 792, -312, 24, 0, 720, -6840, 14496, -10872, 2736, -120, 0, 5040, -86400, 285840, -347904, 171504, -28800, 720, 0, 40320, -1244880, 6181920, -11245680, 8996544, -3090960, 355680, -5040, 0

Offset: 0

Peter Luschny, Aug 11 2022

The Bernoulli numbers with B(1) = 1/2 can be represented as the weighted sum of Eulerian numbers, where we use the definition as given by Graham et al., Eulerian(n, k) = A173018(n, k). For n >= 0 we have
   B_(n) = (1/(n + 1)) * Sum_{k=0..n} (-1)^k * Eulerian(n, k) / binomial(n, k).
The formula was given by Worpitsky in 1883 (see link) as an example for the application of a formula of Schlömilch from 1856. In 2019 the identity was proved in the modern fashion by Gessel on MathOverflow.
For a variant of this identity see the first formula in A356546.
An analogous representation based on the Eulerian numbers of second order is given in A356547.

			The table T(n, k) of the coefficients, sorted in ascending order, starts:
[0]     1;
[1]     1,        0;
[2]     2,       -1,       0;
[3]     6,       -8,       2,         0;
[4]    24,      -66,      44,        -6,       0;
[5]   120,     -624,     792,      -312,      24,        0;
[6]   720,    -6840,   14496,    -10872,    2736,     -120,      0;
[7]  5040,   -86400,  285840,   -347904,  171504,   -28800,    720,     0;
[8] 40320, -1244880, 6181920, -11245680, 8996544, -3090960, 355680, -5040, 0;

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, 2nd ed. Addison-Wesley, Reading, MA, 1994, p. 268. (Since the thirty-fourth printing, Jan. 2022, with B(1) = 1/2.)

Ira Gessel, Eulerian number identity, MathOverflow, Apr 2019.
Peter Luschny, How are the Eulerian numbers of the first-order related to the Eulerian numbers of the second-order?, MathOverflow, Feb. 2021.
Peter Luschny, Eulerian polynomials.
Oskar Schlömilch, Ueber die Bernoulli'sche Funktion und deren Gebrauch bei der Entwickelung halbconvergenter Reihen, Zeitschrift fuer Mathematik und Pysik, vol. 1 (1856), p. 193-211.
Julius Worpitsky, Studien über die Bernoullischen und Eulerschen Zahlen, Journal für die reine und angewandte Mathematik (Crelle), 94 (1883), 203-232. See page 22, first formula.

Cf. A173018 (Eulerian number), A164555(n)/A027642(n) (Bernoulli numbers with B(1) = 1/2), A129814 (row sums, but different sign for n = 1).

Cf. A098361, A123125, A356546, A356547, A356601, A356602.

E1 := proc(n, k) combinat:-eulerian1(n, k) end:
p := (n, x) -> add(E1(n, k)*k!*(n - k)!*(-x)^k, k = 0..n):
seq(print(seq(coeff(p(n, x), x, k), k=0..n)), n = 0..8);
seq(p(n, 1)/(n + 1)!, n = 0..14); # check the Bernoulli representation

T[n_, k_] := k! * (n-k)! * Sum[(-1)^(k-j) * (k-j+1)^n * Binomial[n+1, j], {j, 0, k}]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // TableForm
(* Diagonals: *)
d[n_, k_] := k! * (n - k)! * Sum[(-1)^(n-k-j)*(n - j - k + 1)^n * Binomial[n + 1, j], {j, 0, n - k}];

Let p_n(x) = Sum_{k=0..n} Eulerian(n, k)*k!*(n - k)! * (-x)^k. For x = 1 these polynomials give rise to the representation Bernoulli(n) = p_n(1) / (n + 1)!.
T(n, k) = [x^k] p_n(x).
T(n, k) = (-1)^k*Eulerian(n, k)*k!*(n - k)!.
T(n, k) = k! * (n-k)! * Sum_{j=0..k} (-1)^(k-j)*(k-j+1)^n*binomial(n+1, j).
T(n, k) = (-1)^k * A173018(n, k) * A098361(n, k).
T(n, k) = (-1)^k * A123125(n, n - k) * A098361(n, n - k).

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.

Original entry on oeis.org

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

Peter Luschny, Aug 15 2022

			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;

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

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

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;

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

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)).

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.

Original entry on oeis.org

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

Peter Luschny, Aug 15 2022

			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;

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

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;

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

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)).

cp's OEIS Frontend

A356545 Triangle read by rows. T(n, k) are the coefficients of polynomials p_n(x) based on the Eulerian numbers of first order representing the Bernoulli numbers as B_n = p_n(1) / (n + 1)!.

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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.

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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.

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cp's OEIS Frontend

A356545 Triangle read by rows. T(n, k) are the coefficients of polynomials p_n(x) based on the Eulerian numbers of first order representing the Bernoulli numbers as B_n = p_n(1) / (n + 1)!.

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Programs

Maple

Mathematica

Formula

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.

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Examples

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Programs

Maple

Mathematica

Formula

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.

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Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

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.

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.