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

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

Author

Wolfdieter Lang, Aug 06 2017

Keywords

• sign
• easy
• frac
• tabl

Comments

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.)

Examples

			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.
		

References

• 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.

Links

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

Crossrefs

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

Formula

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).