This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A075513 #70 May 12 2025 12:00:29
%S A075513 1,-1,2,1,-8,9,-1,24,-81,64,1,-64,486,-1024,625,-1,160,-2430,10240,
%T A075513 -15625,7776,1,-384,10935,-81920,234375,-279936,117649,-1,896,-45927,
%U A075513 573440,-2734375,5878656,-5764801,2097152,1,-2048,183708,-3670016,27343750,-94058496,161414428,-134217728,43046721
%N A075513 Triangle read by rows. T(n, m) are the coefficients of Sidi polynomials.
%C A075513 Coefficients of the Sidi polynomials (-1)^(n-1)*D_{n-1,1,n-1}(x), for n >=1, where D_{k,n,m}(z) is given in Theorem 4.2., p. 862, of Sidi [1980].
%C A075513 The row polynomials p(n, x) := Sum_{m=0..n-1} a(n, m)x^m, n >= 1, are obtained from ((Eu(x)^n)*(x-1)^n)/(n*x), where Eu(x) := xd/dx is the Euler-derivative with respect to x.
%C A075513 The row polynomials p(n, y) := Sum_{m=0..n-1} a(n, m)*y^m, n >= 1, are also obtained from ((d^m/dx^m)((exp(x)-1)^m)/m)/exp(x) after replacement of exp(x) by y. Here (d^m/dx^m)f(x), m >= 1, denotes m-fold differentiation of f(x) with respect to x.
%C A075513 b(k,m,n) := (Sum_{p=0..m-1} (a(m, p)*((p+1)*k)^n))/(m-1)!, n >= 0, has g.f. 1/Product_{p=1..m} (1 - k*p*x) for k = 1, 2,... and m = 1, 2,...
%C A075513 The (signed) row sums give A000142(n-1), n >= 1, (factorials) and (unsigned) A074932(n).
%C A075513 The (unsigned) columns give A000012 (powers of 1), 2*A001787(n+1), (3^2)*A027472(n), (4^3)*A038846(n-1), (5^4)*A036071(n-5), (6^5)*A036084(n-6), (7^6)*A036226(n-7), (8^7)*A053107(n-8) for m=0..7.
%C A075513 Right edge of triangle is A000169. - _Michel Marcus_, May 17 2013
%D A075513 A. Sidi, Practical Extrapolation Methods: Theory and Applications, Cambridge University Press, Cambridge, 2003.
%H A075513 Wolfdieter Lang, <a href="/A075513/a075513.pdf">On a Certain Family of Sidi Polynomials</a>, May 2023.
%H A075513 Harlan J. Brothers, <a href="https://doi.org/10.35834/2025/3701067">Pascal's triangle, Sidi polynomials, and powers of e</a>, Missouri J. Math. Sci. (2025) Vol. 37, No. 1, 67-78.
%H A075513 Doron S. Lubinsky and Herbert Stahl, <a href="http://people.math.gatech.edu/~lubinsky/Research%20papers/GatlinburgProcRohrsVn.pdf">Some Explicit Biorthogonal Polynomials</a>, (IN) Approximation Theory XI, (C.K. Chui, M. Neamtu, L. Schumaker, eds.), Nashboro Press, Nashville, 2005, pp. 279-285.
%H A075513 Avram Sidi, <a href="https://doi.org/10.1090/S0025-5718-1980-0572861-2">Numerical Quadrature and Nonlinear Sequence Transformations; Unified Rules for Efficient Computation of Integrals with Algebraic and Logarithmic Endpoint Singularities</a>, Math. Comp., 35 (1980), 851-874.
%F A075513 T(n, m) = ((-1)^(n-m-1)) binomial(n-1, m)*(m+1)^(n-1), n >= m+1 >= 1, else 0.
%F A075513 G.f. for m-th column: ((m+1)^m)(x/(1+(m+1)*x))^(m+1), m >= 0.
%F A075513 E.g.f.: -LambertW(-x*y*exp(-x))/((1+LambertW(-x*y*exp(-x)))*x*y). - _Vladeta Jovovic_, Feb 13 2008 [corrected for offset 0 <= m <= n. For offset n >= 1 take the integral over x. - _Wolfdieter Lang_, Oct 12 2022]
%F A075513 T(n, k) = S(n, k+1) / n where S(, ) is triangle in A258773. - _Michael Somos_, May 13 2018
%F A075513 E.g.f. of column k, with offset n >= 0: exp(-(k + 1)*x)*((k + 1)*x)^k/k!. - _Wolfdieter Lang_, Oct 20 2022
%F A075513 E.g.f: 1/(exp(LambertW(-exp(-x)*x*y) + x) - x*y) assuming offset = 0. - _Peter Luschny_, Oct 21 2022
%e A075513 The triangle T(n, m) begins:
%e A075513 n\m 0 1 2 3 4 5 6 7 8
%e A075513 1: 1
%e A075513 2: -1 2
%e A075513 3: 1 -8 9
%e A075513 4: -1 24 -81 64
%e A075513 5: 1 -64 486 -1024 625
%e A075513 6: -1 160 -2430 10240 -15625 7776
%e A075513 7: 1 -384 10935 -81920 234375 -279936 117649
%e A075513 8: -1 896 -45927 573440 -2734375 5878656 -5764801 2097152
%e A075513 9: 1 -2048 183708 -3670016 27343750 -94058496 161414428 -134217728 4304672
%e A075513 [Reformatted by _Wolfdieter Lang_, Oct 12 2022]
%e A075513 -----------------------------------------------------------------------------
%e A075513 p(2,x) = -1+2*x = (1/(2*x))*x*(d/dx)*x*(d/dx)*(x-1)^2.
%p A075513 # Assuming offset 0.
%p A075513 seq(seq((-1)^(n-k)*binomial(n, k)*(k+1)^n, k=0..n), n=0..8);
%p A075513 # Alternative:
%p A075513 egf := x -> 1/(exp(LambertW(-exp(-x)*x*y) + x) - x*y):
%p A075513 ser := x -> series(egf(x), x, 12):
%p A075513 row := n -> seq(coeff(n!*coeff(ser(x), x, n), y, k), k=0..n):
%p A075513 seq(print(row(n)), n = 0..8); # _Peter Luschny_, Oct 21 2022
%t A075513 p[n_, x_] := p[n, x] = Nest[ x*D[#, x]& , (x-1)^n, n]/(n*x); a[n_, m_] := Coefficient[ p[n, x], x, m]; Table[a[n, m], {n, 1, 9}, {m, 0, n-1}] // Flatten (* _Jean-François Alcover_, Jul 03 2013 *)
%o A075513 (PARI) tabl(nn) = {for (n=1, nn, for (m=0, n-1, print1((-1)^(n-m-1)*binomial(n-1, m)*(m+1)^(n-1), ", ");); print(););} \\ _Michel Marcus_, May 17 2013
%Y A075513 Cf. A075510, A075511, A075512, A074932, A075515, A075516, A075906..A075925, A076002..A076013, A258773.
%K A075513 sign,tabl,easy
%O A075513 1,3
%A A075513 _Wolfdieter Lang_, Oct 02 2002