A051717 -id:A051717 - cp's OEIS frontend

A129826 Transformed Bernoulli twin numbers.

1, -1, -2, -4, -4, 24, 120, -960, -12096, 120960, 3024000, -36288000, -1576143360, 22066007040, 1525620096000, -24409921536000, -2522591034163200, 45406638614937600, 6686974460694528000, -133739489213890560000, -27033456071346536448000, 594736033569623801856000

Offset: 0

Paul Curtz, May 20 2007

sign

G. C. Greubel, Table of n, a(n) for n = 0..300

Cf. A027641, A027642, A051717, A051718, A129378, A140351.

f:= func< n | n le 2 select (-1)^Floor((n+1)/2)/(n+1) else (-1)^n*BernoulliNumber(Floor(n - (1-(-1)^n)/2)) >;
A129826:= func< n | Factorial(n+1)*f(n) >;
[A129826(n): n in [0..30]]; // G. C. Greubel, Feb 01 2024

c[n_?EvenQ] := BernoulliB[n]; c[n_?OddQ] := -BernoulliB[n-1]; c[1]=-1/2; c[2]=-1/3; a[n_] := (n+1)!*c[n]; Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Aug 08 2012 *)

def f(n): return (-1)^((n+1)//2)/(n+1) if n<3 else (-1)^n*bernoulli(n-(n%2))
def A129826(n): return factorial(n+1)*f(n)
[A129826(n) for n in range(31)] # G. C. Greubel, Feb 01 2024

We define Bernoulli twin numbers C(n) via Bernoulli numbers B(n) = A027641(n)/A027642(n) as C(0)=1, 2C(1)=-1, 3C(2)=-1, C(2n-1)= -B(2n-2) and C(2n)=B(2n), n>1. The sequence is defined as a(n)=(n+1)!*C(n).
a(n) = (n+1)!*C(n), where C(n) = A051718(n)/A051717(n).
E.g.f.: Sum(n>=0) C(n) x^n/n! = 1 + x - x^2/2 + Sum_{n>=1} (B(n) - B(n-1))*x^n/n! = x - x^2/2 + x/(e^x-1) - Integral_{y=0..x} ((y dy)/(e^y-1)).

Edited and extended by R. J. Mathar, Aug 06 2008

A168516 Table of the numerators of the fractions of Bernoulli twin numbers and their higher-order differences, read by antidiagonals.

Original entry on oeis.org

-1, 1, -1, -1, 2, -1, -1, -1, 1, 1, 1, -1, -8, -1, 1, 1, 1, 4, -4, -1, -1, -1, -1, 4, 8, 4, -1, -1, -1, -1, -8, -4, 4, 8, 1, 1, 5, 7, -4, -116, -32, -116, -4, 7, 5, 5, 5, 32, 28, 16, -16, -28, -32, -5, -5, -691, -2663, -388, 2524, 5072, 6112, 5072, 2524, -388, -2663, -691, -691, -691, -10264, -10652, -8128, -3056, 3056, 8128, 10652, 10264, 691, 691, 7, 1247, 556, -4148, -2960, -22928

Offset: 0

Paul Curtz, Nov 28 2009

Consider the Bernoulli twin numbers C(n) = A051716(n)/A051717(n) in the top row and successive higher order differences in the other rows of an array T(0,k) = C(k), T(n,k) = T(n-1,k+1)-T(n-1,k):
1, -1/2, -1/3, -1/6, -1/30, 1/30, 1/42, -1/42, -1/30, 1/30, 5/66, -5/66, ...
-3/2, 1/6, 1/6, 2/15, 1/15, -1/105, -1/21, -1/105, 1/15, 7/165, -5/33, ...
5/3, 0, -1/30, -1/15, -8/105, -4/105, 4/105, 8/105, -4/165, -32/165, ...
-5/3, -1/30, -1/30, -1/105, 4/105, 8/105, 4/105, -116/1155, -28/165, ...
49/30, 0, 1/42, 1/21, 4/105, -4/105, -32/231, -16/231, 5072/15015, 8128/15015, ...
-49/30, 1/42, 1/42, -1/105, -8/105, -116/1155, 16/231, 6112/15015, ...
Remove the two leftmost columns:
-1/3, -1/6, -1/30,  1/30, 1/42, -1/42, -1/30, 1/30, 5/66, -5/66,-691/2730, 691/2730, ...
1/6, 2/15, 1/15, -1/105, -1/21, -1/105, 1/15, 7/165, -5/33, -2663/15015, 691/1365, ...
-1/30, -1/15, -8/105, -4/105, 4/105, 8/105, -4/165, -32/165, -388/15015, 10264/15015, ...
-1/30, -1/105, 4/105, 8/105, 4/105, -116/1155, -28/165, 2524/15015, ...
1/42, 1/21, 4/105, -4/105, -32/231, -16/231, 5072/15015, 8128/15015, -2960/3003, ...
1/42, -1/105, -8/105, -116/1155, 16/231, 6112/15015, 3056/15015, -22928/15015, -7184/3003, ...
-1/30, -1/15, -4/165, 28/165, 5072/15015, -3056/15015, -3712/2145, ...
-1/30, 7/165, 32/165, 2524/15015, -8128/15015, -22928/15015, ...
and read the numerators upwards along antidiagonals to obtain the current sequence.
The leftmost column (i.e., the inverse binomial transform of the top row) in this chopped variant equals the top row up to a sign pattern (-1)^n.
In that sense, the C(n) with n>=2 are an eigensequence of the inverse binomial transform (i.e., an autosequence).

Cf. A168426 (denominators), A085737, A085738.

C := proc(n) if n=0 then 1; elif n mod 2 = 0 then bernoulli(n)+bernoulli(n-1); else -bernoulli(n)-bernoulli(n-1); end if; end proc:
A168516 := proc(n,k) L := [seq(C(i),i=0..n+k+3)] ; for c from 1 to n do L := DIFF(L) ; end do; numer(op(k+3,L)) ; end proc:
for d from 0 to 15 do for k from 0 to d do printf("%a,",A168516(d-k,k)) ; end do: end do: # R. J. Mathar, Jul 10 2011

max = 13; c[0] = 1; c[n_?EvenQ] := BernoulliB[n] + BernoulliB[n-1]; c[n_?OddQ] := -BernoulliB[n] - BernoulliB[n-1]; cc = Table[c[n], {n, 0, max+1}]; diff = Drop[#, 2]& /@ Table[ Differences[cc, n], {n, 0, max-1}]; Flatten[ Table[ diff[[n-k+1, k]], {n, 1, max}, {k, 1, n}]] // Numerator (* Jean-François Alcover, Aug 09 2012 *)

Edited and extended by R. J. Mathar, Jul 10 2011

A168426 Square array of denominators of a truncated array of Bernoulli twin numbers (A168516), read by antidiagonals.

Original entry on oeis.org

3, 6, 6, 30, 15, 30, 30, 15, 15, 30, 42, 105, 105, 105, 42, 42, 21, 105, 105, 21, 42, 30, 105, 105, 105, 105, 105, 30, 30, 15, 105, 105, 105, 105, 15, 30, 66, 165, 165, 1155, 231, 1155, 165, 165, 66, 66, 33, 165, 165, 231, 231, 165, 165, 33, 66, 2730, 15015, 15015, 15015, 15015, 15015, 15015, 15015

Offset: 0

Paul Curtz, Nov 25 2009

Entries are multiples of 3.
The sequence of fractions A051716()/A051717() is a sequence of first differences of A164555()/A027642().
It can be observed (see the difference array in A190339) that A168516/A168426 is a sequence of autosequences of the second kind. - Paul Curtz, Dec 21 2016

Cf. A085738, A085737, A168516, A190339, A240581/A239315.

max = 11; c[0] = 1; c[n_?EvenQ] := BernoulliB[n] + BernoulliB[n-1]; c[n_?OddQ] := -BernoulliB[n] - BernoulliB[n-1]; cc = Table[c[n], {n, 0, max+1}]; diff = Drop[#, 2]& /@ Table[ Differences[cc, n], {n, 0, max-1}]; Flatten[ Table[ diff[[n-k+1, k]], {n, 1, max}, {k, 1, n}]] // Denominator (* Jean-François Alcover, Aug 09 2012 *)

More terms from R. J. Mathar, Jul 10 2011

A239315 Array read by antidiagonals: denominators of the core of the classical Bernoulli numbers.

Original entry on oeis.org

15, 15, 15, 105, 105, 105, 21, 105, 105, 21, 105, 105, 105, 105, 105, 15, 105, 105, 105, 105, 15, 165, 165, 1155, 231, 1155, 165, 165, 33, 165, 165, 231, 231, 165, 165, 33, 15015, 15015, 15015, 15015, 15015, 15015, 15015, 15015, 15015

Offset: 0

Paul Curtz, Mar 15 2014

We consider the autosequence A164555(n)/A027642(n) (see A190339(n)) and its difference table without the first two rows and the first two columns:
2/15,     1/15,    -1/105,   -1/21,   -1/105,     1/15,   7/165,   -5/33,...
-1/15,  -8/105,    -4/105,   4/105,    8/105,   -4/165, -32/165,...
-1/105,  4/105,     8/105,   4/105, -116/1155, -28/165,...
1/21,    4/105,    -4/105, -32/231,   -16/231,...
-1/105, -8/105, -116/1155,  16/231,...
-1/15,  -4/165,    28/165,...
7/165,  32/165,...
5/33,... etc.
This is an autosequence of the second kind.
The antidiagonals are palindromes in absolute values.
a(n) are the denominators. Multiples of 3.
Sum of odd antidiagonals: 2/15, -2/21, 2/15, -10/33, 1382/1365,... = -2*A000367(n+2)/A001897(n+2).
The sum of the even antidiagonals is A000004.
2/15, 0, -2/21,... = -4*A027641(n+4)/A027642(n+4) = -4*A164555(n)/A027642(n+4) and others.

			As a triangle:
15,
15,   15,
105, 105, 105,
21,  105, 105, 21,
105, 105, 105, 105, 105,
etc.

Cf. A085737/A085738, A168516/A168426 (autosequence), A027641, A176327/A176289, A235774, A165161/A051717(n+1).

max = 12; tb = Table[BernoulliB[n], {n, 0, max}]; td = Table[Differences[tb, n][[3 ;; -1]], {n, 2, max - 1}]; Table[td[[n - k + 1, k]] // Denominator, {n, 1, max - 3}, {k, 1, n}] // Flatten (* Jean-François Alcover, Apr 11 2014 *)

A129724 a(0) = 1; then a(n) = n!(1 - (-1)^nBernoulli(n-1)).

Original entry on oeis.org

1, 2, 3, 7, 24, 116, 720, 5160, 40320, 350784, 3628800, 42940800, 479001600, 4650877440, 87178291200, 2833294464000, 20922789888000, -2166903606067200, 6402373705728000, 6808619561103360000, 2432902008176640000, -26982365129174827008000, 1124000727777607680000

Offset: 0

Paul Curtz, Jun 02 2007

sign

G. C. Greubel, Table of n, a(n) for n = 0..300

Cf. A027641, A027642, A051716, A051717, A129825, A129826.

Concatenation([1], List([1..25], n-> Factorial(n)*(1 - (-1)^n*Bernoulli(n-1)) )); # G. C. Greubel, Dec 03 2019

[n eq 0 select 1 else Factorial(n)*(1 - (-1)^n*Bernoulli(n-1)): n in [0..25]]; // G. C. Greubel, Dec 03 2019

a:= proc(n)
      if n=0 and n>=0 then 1
    elif n mod 2 = 0 then n!*(1 - bernoulli(n-1))
    else n!*(1 + bernoulli(n-1))
      fi; end;
seq(a(n), n=0..25); # modified by G. C. Greubel, Dec 03 2019

a[0] = 1; a[n_]:= n!*(1-(-1)^n*BernoulliB[n-1]); Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Sep 16 2013 *)

a(n) = if(n==0, 1, n!*(1 - (-1)^n*bernfrac(n-1)) ); \\ G. C. Greubel, Dec 03 2019

[1]+[factorial(n)*(1 - (-1)^n*bernoulli(n-1)) for n in (1..25)] # G. C. Greubel, Dec 03 2019

Edited with simpler definition by N. J. A. Sloane, May 25 2008

A172083 1, followed by numerators of first differences of Bernoulli numbers (B(i) - B(i-1)).

Original entry on oeis.org

1, -3, 2, -1, -1, 1, 1, -1, -1, 1, 5, -5, -691, 691, 7, -7, -3617, 3617, 43867, -43867, -174611, 174611, 854513, -854513, -236364091, 236364091, 8553103, -8553103, -23749461029, 23749461029, 8615841276005, -8615841276005, -7709321041217, 7709321041217, 2577687858367

Offset: 0

Paul Curtz, Jan 25 2010

sign

			Bernoulli numbers: 1, -1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, ...
First differences: -3/2, 2/3, -1/6, -1/30, 1/30, 1/42, -1/42, -1/30, ...
Numerators: -3, 2, -1, -1, 1, 1, -1, -1, 1, 5, -5, -691, 691, 7, ...
Denominators: 2, 3, 6, 30, 30, 42, 42, 30, 30, 66, 66, 2730, ...

Cf. A027641/A027642.

For denominators see A051717.

Edited and more terms from N. J. A. Sloane, Apr 22 2021

A140333 Triangle T(n,k) with the coefficient [x^k] (n+1)!* C(n,x), in row n, column k, where C(.,.) are the Bernoulli twin number polynomials of A129378.

Original entry on oeis.org

1, -1, 2, -2, 0, 6, -4, -12, 12, 24, -4, -60, -60, 120, 120, 24, -120, -720, -240, 1080, 720, 120, 840, -2520, -8400, 0, 10080, 5040, -960, 6720, 20160, -47040, -100800, 20160, 100800, 40320, -12096, -60480, 241920, 423360

Offset: 0

Paul Curtz, May 28 2008

The terms at x=0 define the Bernoulli twin numbers, C(n,0)=C(n) = A129826(n)/(n+1)! .
Because the C(n,x) are derived from the Bernoulli polynomials B(n,x) via a binomial transformation and because the odd-indexed Bernoulli numbers are (essentially) zero, the following sum rules for the C(n) emerge (partially in Umbral notation):
For odd C(n): C(2n)=(C-1)^(2n-1), n > 1, C(2n) disappears; example: C(4)=C(4)-3C(3)+3C(2)-C(1).
0r for C(2n+1): (C-1)^2n=0, n >0; example: C(1)-4C(2)+6C(3)-4C(4)+C(5)=0.
With positive coefficients, table
1, 2;
2, 2, 3;
3, 2, 3, 6;
4, 2, 3, 6, 30;
5, 2, 3, 6, 30, -30;
6, 2, 3, 6, 30, -30, -42;
gives C(n). Example: 3C(0)+2C(1)+3C(2)+6C(3)=0. See -A051717(n+1), Bernoulli twin numbers denominators, with from 30 opposite twin.

			1;    C(0,x) = 1
-1, 2;    C(1,x) = -1/2+x
-2, 0, 6;       C(2,x) = -1/3+x^2
-4, -12, 12, 24;      C(3,x) = -1/6 -x/2 +x^2/2 +x^3
-4, -60, -60, 120, 120;

Cf. A129378.

C := proc(n,x) if n =0 then 1; else add( binomial(n-1,j-1)*bernoulli(j,x),j=1..n) ; expand(%) ; end if; end proc:
A140333 := proc(n,k) (n+1)!*C(n,x) ; coeftayl(%,x=0,k) ; end proc: # R. J. Mathar, Jun 27 2011

c[0, ] = 1; c[n, x_] := Sum[Binomial[n-1, j-1]*BernoulliB[j, x], {j, 1, n}]; t[n_, k_] := (n+1)!*Coefficient[c[n, x], x, k]; Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 16 2013 *)

A162508 A binomial sum of powers related to the Bernoulli numbers, triangular array, read by rows.

Original entry on oeis.org

-1, -2, 2, -4, 10, -6, -8, 38, -54, 24, -16, 130, -330, 336, -120, -32, 422, -1710, 3000, -2400, 720, -64, 1330, -8106, 21840, -29400, 19440, -5040, -128, 4118, -36414, 141624, -285600, 312480, -176400, 40320

Offset: 1

Peter Luschny, Jul 05 2009

T(n,k) = sum_{v=0..k} (-1)^v*v*binomial(k,v)*(v+1)^(n-1)
for n >= 1, k >= 1; by convention T(0,0) = 1.
Gives a representation of the Bernoulli numbers B_{n} = B_{n}(1) (with B_1 = 1/2)
B_{n} = sum_{j=0..n} sum_{k=0..j} T(j,k)/(k+1)
T(n,1) = -2^(n-1) (n>=1)
T(n,n) = (-1)^n*n! (n>=1)
sum_{k=0..n} T(n,k) = -A000007(n-1) = -1,0,0,0,0,... (n>=1)
sum_{k=0..n} abs(T(n,k)) = A162509(n) = A073146(n,n-1) (n>=1)
sum_{k=0..n} T(n,k)/(k+1) = Bernoulli(n,1)-Bernoulli(n-1,1) (n>=1)
numer(sum(T(n,k)/(k+1),k=0..n)) = A051716(n) (n>=0)
denom(sum(T(n,k)/(k+1),k=0..n)) = A051717(n) (n>=0)
Contribution from Peter Luschny, Jul 08 2009: (Start)
More generally, define the polynomials (assume p[0,0](x)=1 and 0^0=1)
p[n,k](x) = sum_{v=0..k} (-1)^v*v*binomial(k,v)*(v+1+x)^(n-1)
[1], [0, -1], [0, -2-x, 2], [0, -4-4x-x^2, 10+4x, -6], ...
then T(n,k)=p[n,k](0) and (-1)^k*k!*Stirling2(n,k)=p[n,k](-1) (cf. A019538).
Assume now k >= 1 and read by rows. Then
p[n,k](1) = -1,-3,2,-9,14,-6,-27,74,-72,24,-81,350,-582,432,-120,...
(-1)^n*(-2)^(n-k)*p[n,k](-1/2))=1,3,2,9,16,6,27,98,90,24,81,544,924,576,120,..
(-1)^n*(-2)^(n-k)*p[n,k](-3/2))=1,1,2,1,8,6,1,26,54,24,1,80,348,384,120,... (End)
Variant of A199400.

			For n >= 1, k >= 1:
..................... -1
................... -2, 2
................. -4, 10, -6
.............. -8, 38, -54, 24
......... -16, 130, -330, 336, -120
..... -32, 422, -1710, 3000, -2400, 720
-64, 1330, -8106, 21840, -29400, 19440, -5040

Vincenzo Librandi, Rows n = 1..50, flattened

Cf. A162509, A073146, A051716, A051717, A199400. A019538, A028246, A038719.

T := proc(n,k) local v; if n=0 and k=0 then 1 else
add((-1)^v*v*binomial(k,v)*(v+1)^(n-1),v=0..k) fi end:
# Peter Bala's e.g.f. assuming offset 0:
egf := (x, z) -> -((1-x)/exp(z) + x)^(-2):
ser := series(egf(x, z), z, 10): coz := n -> n!*coeff(ser, z, n):
row := n -> seq(coeff(coz(n), x, k), k = 0..n):
seq(print(row(n)), n = 0..9); # Peter Luschny, Jan 28 2021

t[n_, k_] := Sum[(-1)^v*v*Binomial[k, v]*(v + 1)^(n - 1), {v, 0, k}]; Table[t[n, k], {n, 1, 8}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 26 2013, after Maple *)

def A162508(n, k):
    if n==0 and k==0: return 1
    return add((-1)^v*v*binomial(k, v)*(v+1)^(n-1) for v in (0..k))
for n in (1..8): [A162508(n, k) for k in (1..n)] # Peter Luschny, Jul 21 2014

From Peter Bala, Jul 21 2014: (Start)
T(n,k) = (-1)^k*k!*( Stirling2(n+1,k+1) - Stirling2(n,k+1) ), 1 <= k <= n.
T(n,k) = (-1)^k*(k + 1)*A038719(n+1,k+1).
E.g.f.: - B(-x,z)^2, where B(x,z) = 1/((1 + x)*exp(-z) - x) = 1 + (1 + x)*z + (1 + 3*x + 2*x^2)*z^2/2! + ... is an e.g.f. for A028246 (with an offset of 0).
Recurrence: T(n,k) = (k + 1)*T(n-1,k) - k*T(n-1,k-1).
The unsigned version of the triangle equals the matrix product A007318*A019538.
Assuming this triangle is a signed version of A199400 then the n-th row polynomial R(n,x) = 1/(1 - x)*( sum {k = 1..inf} k*(k + 1)^(n-1)*(x/(x - 1))^k ), valid for x in the open interval (-inf, 1/2). (End)

More terms from Philippe Deléham, Nov 05 2011

A235774 Let b(k) = A164555(k)/A027642(k), the sequence of "original" Bernoulli numbers with -1 instead of A164555(0)=1; then a(n) = numerator of the n-th term of the binomial transform of the b(k) sequence.

Original entry on oeis.org

-1, -1, 1, 1, 59, 3, 169, 5, 179, 7, 533, 9, 26609, 11, 79, 13, 3523, 15, 56635, 17, -168671, 19, 857273, 21, -236304031, 23, 8553247, 25, -23749438409, 27, 8615841677021, 29, -7709321025917, 31, 2577687858559, 33, -26315271552988224913

Offset: 0

Paul Curtz, Jan 15 2014

(a(n)/A027642(n)) = -1, -1/2, 1/6, 1, 59/30, 3, 169/42, 5, 179/30, 7, 533/66, 9,.. .
Difference table for a(n)/A027642(n):
-1, -1/2,   1/6,      1,  59/30,     3, 169/42, ...
1/2, 2/3,   5/6,  29/30,  31/30, 43/42,  41/42, ... = A165161(n)/A051717(n+1)
1/6, 1/6,  2/15,   1/15, -1/105, -1/21, -1/105, ... not in the OEIS
0, -1/30, -1/15, -8/105, -4/105, 4/105,  8/105, ... etc.
Compare with the array in A190339.

Cf. A164558/A027642, A165161(n)/A051717(n+1).

b[0] = -1; b[1] = 1/2; b[n_] := BernoulliB[n]; a[n_] := Sum[Binomial[n, k]*b[k], {k, 0, n}] // Numerator; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jan 30 2014 *)

(a(n+1) - a(n))/A027642(n) = A165161(n)/A051717(n+1).
(A164558(n) - a(n))/A027642(n) = 2's = A007395.
(a(n) - A164555(n))/A027642(n) = n - 2 = A023444(n).

A290317 Triangle read by rows. Row n gives the numerators of the coefficients of the Bernoulli polynomials of the second kind (in rising powers).

Original entry on oeis.org

1, 1, 1, -1, 0, 1, 1, 0, -3, 1, -19, 0, 4, -4, 1, 9, 0, -15, 55, -15, 1, -863, 0, 72, -100, 105, -12, 1, 1375, 0, -420, 1918, -1575, 119, -35, 1, -33953, 0, 2880, -4704, 3248, -1176, 700, -24, 1, 57281, 0, -22680, 39204, -29547, 60921, -2940, 414, -63, 1, -3250433, 0, 201600, -365280, 295310, -134568, 37415, -6480, 1365, -40, 1

Offset: 0

Wolfdieter Lang, Aug 06 2017

For the denominators see A290318.
See the Weisstein link and the Roman reference for Bernoulli polynomials of the second kind.
The Bernoulli polynomials of the second kind B2(n, x) = Sum_{k=0..n} r(n, k)*x^k, with the rationals r(n, k) = T(n, k)/A290318(n, k), are the Sheffer polynomials (t/log(1 + t), log(1 + t)) (this notation differs from Roman's one). B2(n, x) = [t^n/n!] (t*(1 + t)^x / log(1 + t)). This means that the e.g.f of the sequence of column k (with leading zeros) is t*(log(1 + t))^(k-1)/k!, for k >= 0.
The rational triangle r(n, k) multiplied by A002790(n) becomes an integer triangle looking like  A157982.
The a-sequence for the Sheffer polynomials B2(n, x) has e.g.f. t/(exp(t) - 1). aB2(n) = B_n = A027641(n) / A027642(n). The z-sequence has e.g.f. (exp(t) - (1+t))/(1 - exp(x))^2, with zB2(n) = (-1)^(n+1)*A051716(n+1) / A051717(n+1)
(n+1). (For a- and z-sequences of Sheffer triangles see the W. Lang link with references in A006232.)

			The triangle T(n, k) begins:
n\k         0 1      2       3      4       5     6     7    8   9  10 ...
0:          1
1:          1 1
2:         -1 0      1
3:          1 0     -3       1
4:        -19 0      4      -4      1
5:          9 0    -15      55    -15       1
6:       -863 0     72    -100    105     -12     1
7:       1375 0   -420    1918  -1575     119   -35     1
8:     -33953 0   2880   -4704   3248   -1176   700   -24    1
9:      57281 0 -22680   39204 -29547   60921 -2940   414  -63   1
10:  -3250433 0 201600 -365280 295310 -134568 37415 -6480 1365 -40   1
...
--------------------------------------------------------------------------------
The triangle of the rationals r(n, k) = T(n, k)/A290318(n, k) begins:
n\k             0  1      2       3        4       5      6     7      8   9 10
0:              1
1:            1/2  1
2:           -1/6  0      1
3:            1/4  0   -3/2       1
4:          19/30  0      4      -4        1
5:            9/4  0    -15    55/3    -15/2       1
6:        -863/84  0     72    -100    105/2     -12      1
7:        1375/24  0   -420  1918/3  -1575/4     119  -35/2
8:      -33953/90  0   2880   -4704     3248   -1176  700/3   -24      1
9:       57281/20  0 -22680   39204   -29547 60921/5  -2940   414  -63/2   1
10:  -3250433/132  0 201600 -365280   295310 -134568  37415 -6480 1365/2 -40  1
...
The first polynomials B2(n, x) are:
B2(0, x) =   1,
B2(1, x) =  1/2 + x,
B2(2, x) = -1/6     + x^2,
B2(3, x) =  1/4     - (3/2)*x^2 + x^3,
...
Recurrence from Sheffer a- and z-sequence:
r(3, 0) = 3*((1/2)*r(2,0) + (-1/3)*r(2,1) + (1/6)*r(2, 2)) = 3*(-1/12  + 0 + 1/6) = 1/4.
r(4, 2) = (4/2)*(1*1*r(3, 1) + 2*(-1/2)*r(3, 2) + 3*(1/6)*r(3, 3)) = 2*(0 - (-3/2) + 1/2) = 4.
General Sheffer recurrence for B2(n, x): B2(3, x) = x*B2(2, x-1) +
F(2, d_x)*B2(2, x) = ((5/6)*x - 2*x^2 + x^3) + (1/2 + (-5/12)*d/dx + (1/3)*(1/2!)*d^2/dx^2)*(-1/6+ x^2) = 1/4 - (3/2)*x^2 + x^3.The rationals s(n) begin {1/2, -5/12, 1/3, -31/120, 1/5, -41/252,  ...}.
Boas-Buck identity for B2(3, x) check: (x*d/dx - 3*1)(1/4 - (3/2)*x^2 + x^3) - 3!*(x*d/dx - 1)* *((1/2)*B2(2, x)/2! + (-5/12)*B2(1, x)/1! + (3/8)) =  0.
  The alpha sequence begins {1/2, -5/12, 3/8, -251/720, 95/288, -19087/60480, ...}.
Boas-Buck column k = 2 recurrence, for n=2: r(3, 2) = -(3!*1/1)*(1/2!) * alpha(0)*r(2, 2) = -(3!/2!)*(1/2)*1= -3!/4 = -3/2.

Ralph P. Boas, jr. and R. Creighton Buck, Polynomial Expansions of analytic functions, Springer, 1958, pp. 17 - 21, (last sign in eq. (6.11) should be -).
Earl D. Rainville, Special Functions, The Macmillan Company, New York, 1960, ch. 8, sect. 76, 140 - 146.
Steven Roman, The Umbral Calculus, Academic Press,1894, ch. 4, sect. 3.2, pp. 113-119, p. 50, p. 114.

Eric Weisstein's World of Mathematics, Bernoulli Polynomial of the Second Kind.

Cf. A002208/A002209, A002790, A157982, A027641/ A027642, A051716/A051717, A290318 (denominators), A165226(n+1) / A164869(n+1).

T(n, k) = numerator(r(n, k)), with r(n, k) the entries of the rational Sheffer triangle (t/log(1 + t), log(1 + t)) (the coefficients of the Bernoulli polynomials of the second kind).
Recurrence for r(n, k) = T(n, k) / A290318(n, k) from a- and z-sequences (see a comment above): r(0, 0) = 1, r(n, 0) = n*Sum_{j=0..n-1} zB2(j)*r(n-1, j), for n >= 1, and  r(n, m) = (n/k)*Sum_{j=0..n-k} binomial(k-1+j, j)*aB2(j)*r(n-1, k-1+j), with  zB2(n) and aB2(n) given above in a comment.
Meixner type recurrence for monic Sheffer polynomials: B2(n, x + 1) = B2(n, x) + n*B2(n-1, x), B2(0, x) = 1. See Roman, p. 114.
Recurrence for general Sheffer polynomials (see Roman, Corollary 3.7.2, p. 50):
B2(0,x) = 1, B2(n, x) = x*B2(n-1, x-1) + D(n-1, d_x)*B2(n-1, x), for n >= 1 with D(n-1, t) =  Sum_{k=0..n-1} s(k)*t^k/k!, with s(k) = [x^k/k!] ((1-exp(x)*(1-x)) / (x*(exp(x)-1)*exp(x))) and d_x = d/dx. The rationals s(n) = (-1)^n * A165226(n+1) / A164869(n+1).
Boas-Buck identity (see the reference, p.20,  eq. (6.11) (last sign -), and the Rainville reference, p. 141, Theorem 50, computed for the present Shefffer example):
(E_x - n*1)*B2(n, x) - n!*(E_x - 1)*Sum_{k=0..n-1} alpha(k)*B2(n-1-k, x) / (n-1-k)! = 0, for n >= 0, with alpha(k) = A002208(n+1)/A002209(n+1) and E_x = x*d/dx (Euler operator).
Boas-Buck column k recurrence from the preceding identity for the rational Sheffer triangle, for n > k >= 0 with inputs r(k, k) = 1:  r(n, k) = -n!*((k-1)/(n-k))*Sum_{p=k..n-1} (1/p!)*alpha(n-1-p)*r(p, k).

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