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A290317
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Triangle read by rows. Row n gives the numerators of the coefficients of the Bernoulli polynomials of the second kind (in rising powers).
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3
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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
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OFFSET
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0,9
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COMMENTS
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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.)
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REFERENCES
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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.
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LINKS
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FORMULA
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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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EXAMPLE
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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
...
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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.
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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