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

%I A290317 #13 Feb 16 2025 08:33:49
%S A290317 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,
%T A290317 105,-12,1,1375,0,-420,1918,-1575,119,-35,1,-33953,0,2880,-4704,3248,
%U A290317 -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
%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).
%C A290317 For the denominators see A290318.
%C A290317 See the Weisstein link and the Roman reference for Bernoulli polynomials of the second kind.
%C A290317 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.
%C A290317 The rational triangle r(n, k) multiplied by A002790(n) becomes an integer triangle looking like  A157982.
%C A290317 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)
%C A290317 (n+1). (For a- and z-sequences of Sheffer triangles see the W. Lang link with references in A006232.)
%D A290317 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 -).
%D A290317 Earl D. Rainville, Special Functions, The Macmillan Company, New York, 1960, ch. 8, sect. 76, 140 - 146.
%D A290317 Steven Roman, The Umbral Calculus, Academic Press,1894, ch. 4, sect. 3.2, pp. 113-119, p. 50, p. 114.
%H A290317 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BernoulliPolynomialoftheSecondKind.html">Bernoulli Polynomial of the Second Kind</a>.
%F A290317 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).
%F A290317 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.
%F A290317 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.
%F A290317 Recurrence for general Sheffer polynomials (see Roman, Corollary 3.7.2, p. 50):
%F A290317 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).
%F A290317 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):
%F A290317 (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).
%F A290317 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).
%e A290317 The triangle T(n, k) begins:
%e A290317 n\k         0 1      2       3      4       5     6     7    8   9  10 ...
%e A290317 0:          1
%e A290317 1:          1 1
%e A290317 2:         -1 0      1
%e A290317 3:          1 0     -3       1
%e A290317 4:        -19 0      4      -4      1
%e A290317 5:          9 0    -15      55    -15       1
%e A290317 6:       -863 0     72    -100    105     -12     1
%e A290317 7:       1375 0   -420    1918  -1575     119   -35     1
%e A290317 8:     -33953 0   2880   -4704   3248   -1176   700   -24    1
%e A290317 9:      57281 0 -22680   39204 -29547   60921 -2940   414  -63   1
%e A290317 10:  -3250433 0 201600 -365280 295310 -134568 37415 -6480 1365 -40   1
%e A290317 ...
%e A290317 --------------------------------------------------------------------------------
%e A290317 The triangle of the rationals r(n, k) = T(n, k)/A290318(n, k) begins:
%e A290317 n\k             0  1      2       3        4       5      6     7      8   9 10
%e A290317 0:              1
%e A290317 1:            1/2  1
%e A290317 2:           -1/6  0      1
%e A290317 3:            1/4  0   -3/2       1
%e A290317 4:          19/30  0      4      -4        1
%e A290317 5:            9/4  0    -15    55/3    -15/2       1
%e A290317 6:        -863/84  0     72    -100    105/2     -12      1
%e A290317 7:        1375/24  0   -420  1918/3  -1575/4     119  -35/2
%e A290317 8:      -33953/90  0   2880   -4704     3248   -1176  700/3   -24      1
%e A290317 9:       57281/20  0 -22680   39204   -29547 60921/5  -2940   414  -63/2   1
%e A290317 10:  -3250433/132  0 201600 -365280   295310 -134568  37415 -6480 1365/2 -40  1
%e A290317 ...
%e A290317 The first polynomials B2(n, x) are:
%e A290317 B2(0, x) =   1,
%e A290317 B2(1, x) =  1/2 + x,
%e A290317 B2(2, x) = -1/6     + x^2,
%e A290317 B2(3, x) =  1/4     - (3/2)*x^2 + x^3,
%e A290317 ...
%e A290317 Recurrence from Sheffer a- and z-sequence:
%e A290317 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.
%e A290317 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.
%e A290317 General Sheffer recurrence for B2(n, x): B2(3, x) = x*B2(2, x-1) +
%e A290317 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,  ...}.
%e A290317 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.
%e A290317   The alpha sequence begins {1/2, -5/12, 3/8, -251/720, 95/288, -19087/60480, ...}.
%e A290317 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.
%Y A290317 Cf. A002208/A002209, A002790, A157982, A027641/ A027642, A051716/A051717, A290318 (denominators), A165226(n+1) / A164869(n+1).
%K A290317 sign,easy,frac,tabl
%O A290317 0,9
%A A290317 _Wolfdieter Lang_, Aug 06 2017