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 A347781 #40 Mar 22 2025 04:40:39
%S A347781 1,2,-1,-1,1,6,-6,2,-5,8,-3,1,-2,1,24,-36,24,-6,-26,57,-42,11,9,-24,
%T A347781 21,-6,-1,3,-3,1,120,-240,240,-120,24,-154,428,-468,244,-50,71,-236,
%U A347781 294,-164,35,-14,52,-72,44,-10,1,-4,6,-4,1,720,-1800,2400,-1800,720,-120,-1044,3510,-5080,3960,-1620,274,580,-2305,3720,-3070,1300,-225,-155,685,-1210,1070,-475,85,20,-95,180,-170,80,-15,-1,5,-10,10,-5,1
%N A347781 Sequence composed of consecutive square matrices A(d) with dimension d=1,2,3,... Matrix elements are arranged by increasing row index i, and, within fixed i, by increasing column index j. Each block A(d) is related to the inverse of a class of integer Vandermonde matrices.
%C A347781 Square matrices A(d) from the sequence are related to the inverse of Vandermonde matrices of the type V(1-s,...,d-s)[i,j] = (i-s)^(j-1), for 1 <= i,j <= d .
%C A347781 In particular, if s = 0, A(d) = [V(1,...,d)]^(-1) * (d-1)!.
%C A347781 A(d) can be generated using corresponding square blocks in A335442.
%H A347781 ProofWiki, <a href="https://proofwiki.org/wiki/Inverse_of_Vandermonde_Matrix">Inverse of Vandermonde Matrix</a>.
%H A347781 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/K-Function.html">K-Function</a>.
%H A347781 Wikipedia, <a href="https://en.wikipedia.org/wiki/Vandermonde_matrix">Vandermonde matrix</a>.
%F A347781 For d >= 1, if A(d) denotes the d-th square block from the sequence:
%F A347781 .
%F A347781 (1)
%F A347781 If B(d) denotes the corresponding square block from A335442:
%F A347781 A(d)[i,j] = B(d)[d-j+1, d-i+1] * binomial(d-1,j-1) * (-1)^(i+j), for 1 <= i,j <= d
%F A347781 .
%F A347781 (2)
%F A347781 If V(1,...,d) denotes the d-dimensional integer Vandermonde matrix
%F A347781 V(1,...,d)[i,j] = i^(j-1), for 1 <= i,j <= d :
%F A347781 A(d) / (d-1)! = [V(1,...,d)]^(-1) ,
%F A347781 or equivalently, as integer formula:
%F A347781 V(1,...,d) * A(d) = I(d) * (d-1)!
%F A347781 Here, I(d) denotes the d-dimensional identity matrix
%F A347781 .
%F A347781 (3)
%F A347781 More generally, for s = ...,-2,-1,0,1,2,...
%F A347781 If V(1-s,...,d-s) denotes the d-dimensional integer Vandermonde matrix
%F A347781 V(1-s,...,d-s)[i,j] = (i-s)^(j-1), for 1 <= i,j <= d :
%F A347781 T(d,s) * A(d) / (d-1)! = [V(1-s,...,d-s)]^(-1) ,
%F A347781 or equivalently, as integer formula:
%F A347781 V(1-s,...,d-s) * T(d,s) * A(d) = I(d) * (d-1)!
%F A347781 Here, T(d,s) denotes the d-dimensional upper triangular matrix
%F A347781 T(d,s)[i,j] = binomial(j-1,i-1) * s^(j-i) if i <= j
%F A347781 T(d,s)[i,j] = 0 if i > j
%F A347781 .
%F A347781 (4)
%F A347781 determinant[A(d)] = K(d) = A002109(d)
%F A347781 Here, K() denotes the K-function. K(d+1) equals the d-th hyperfactorial.
%F A347781 .
%F A347781 (5)
%F A347781 Row and column sums amount to
%F A347781 Sum_{j=1..d} A(d)[i,j] = delta(i,1) * (d-1)!
%F A347781 Sum_{i=1..d} A(d)[i,j] = delta(j,1) * (d-1)!
%F A347781 Here, delta(i,j) denotes the Kronecker delta.
%e A347781 Matrices begin:
%e A347781 d=1: 1,
%e A347781 .
%e A347781 d=2: 2, -1
%e A347781 -1, 1
%e A347781 .
%e A347781 d=3: 6, -6, 2
%e A347781 -5, 8, -3
%e A347781 1, -2, 1
%e A347781 .
%e A347781 d=4: 24, -36, 24, -6
%e A347781 -26, 57, -42, 11
%e A347781 9, -24, 21, -6
%e A347781 -1, 3, -3, 1 .
%e A347781 .
%e A347781 For example, let d = 3:
%e A347781 .
%e A347781 | 6 -6 2 |
%e A347781 A(3) = | -5 8 -3 |
%e A347781 | 1 -2 1 |
%e A347781 .
%e A347781 | 1 1 1 |
%e A347781 V(1,2,3) = | 1 2 4 |
%e A347781 | 1 3 9 |
%e A347781 .
%e A347781 | 2 0 0 |
%e A347781 V(1,2,3) * A(3) = | 0 2 0 |
%e A347781 | 0 0 2 |
%Y A347781 Cf. A335442, A002109.
%K A347781 sign,tabf
%O A347781 1,2
%A A347781 _Andreas B. G. Blobel_, Sep 13 2021