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 A227343 #15 Feb 16 2025 08:33:20
%S A227343 1,1,1,3,3,1,13,13,6,1,75,75,37,10,1,541,541,270,85,15,1,4683,4683,
%T A227343 2341,770,170,21,1,47293,47293,23646,7861,1890,308,28,1,545835,545835,
%U A227343 272917,90930,22491,4158,518,36,1,7087261,7087261,3543630,1181125,294525,57351,8400,822,45,1
%N A227343 Matrix inverse of triangle A227342.
%C A227343 The e.g.f. has the form A(t)*exp(x*B(t)), where A(t) = 1/(2 - exp(t)) and B(t) = exp(t) - 1. Thus the row polynomials of this triangle form a Sheffer sequence for the pair (1 - t, log(1 + t)) (see Roman, p.17).
%C A227343 Let x_(k) := x*(x-1)*...*(x-k+1) denote the k-th falling factorial polynomial. Define a sequence x_[n] of basis polynomials for the polynomial algebra C[x] by setting x_[0] = 1, and setting x_[n] = x_(n-1)*(x - 2*n + 1) for n >= 1. The sequence begins [1, x-1, x*(x-3), x*(x-1)*(x-5), x*(x-1)*(x-2)*(x-7), ...]. Then this is the triangle of connection constants for expressing the monomial polynomials x^n as a linear combination of the basis x_[k], that is, x^n = sum {k = 0..n} T(n,k)*x_[k]. An example is given below.
%D A227343 S. Roman, The umbral calculus, Pure and Applied Mathematics 111, Academic Press Inc., New York, 1984. Reprinted by Dover in 2005.
%H A227343 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ShefferSequence.html">Sheffer Sequence</a>
%F A227343 E.g.f.: 1/(2 - exp(t))*exp(x*(exp(t) - 1)) = 1 + (1 + x)*t + (3 + 3*x + x^2)*t^2/2! + (13 + 13*x + 6*x^2 + x^3)*t^3/3! + ....
%F A227343 Recurrence equation: T(n,0) = A000670(n), and for k >= 1, T(n,k) = 1/k*sum {i = 1..n} binomial(n,i)*T(n-i,k-1).
%F A227343 The row polynomials R(n,x) satisfy the Sheffer identity R(n,x + y) = sum {k = 0..n} binomial(n,k)*Bell(k,y)*R(n-k,x), where Bell(k,y) is the Bell or exponential polynomial (row polynomials of A048993).
%F A227343 The row polynomials also satisfy d/dx(R(n,x)) = sum {k = 0..n-1} binomial(n,k)*R(k,x).
%F A227343 Row sums A059099. Column 1 and column 2 = A000670. 1 + 2*column 3 = A000670 (apart from the first two terms).
%F A227343 From _Emanuele Munarini_, Dec 21 2016: (Start)
%F A227343 T(n,k) = (n!/k!)*[t^n](exp(t)-1)^k/(2-exp(t)).
%F A227343 T(n,k) = (n!/k!)*[t^(n-k)](t/log(1+t))^(n+1)/(1-t^2). (End)
%e A227343 Triangle begins
%e A227343 n\k| 0 1 2 3 4 5
%e A227343 = = = = = = = = = = = = = = = = =
%e A227343 0 | 1
%e A227343 1 | 1 1
%e A227343 2 | 3 3 1
%e A227343 3 | 13 13 6 1
%e A227343 4 | 75 75 37 10 1
%e A227343 5 | 541 541 270 85 15 1
%e A227343 ...
%e A227343 Connection constants. Row 4 = [75,75,37,10,1]: Thus
%e A227343 75 + 75*(x - 1) + 37*x*(x - 3) + 10*x*(x - 1)*(x - 5)+ x*(x - 1)*(x - 2)*(x - 7) = x^4.
%t A227343 T[n_, k_] := n!/k! SeriesCoefficient[Series[1/(2 - Exp[t]) (Exp[t] - 1)^k, {t, 0, n}], n]
%t A227343 Flatten[Table[T[n, k], {n, 0, 12}, {k, 0, n}]]
%t A227343 U[n_, k_] := n!/k! SeriesCoefficient[Series[1/(1 - t^2) (t/Log[1 + t])^(n + 1), {t, 0, n - k}], n - k]
%t A227343 Flatten[Table[U[n, k], {n, 0, 8}, {k, 0, n}]] (* _Emanuele Munarini_, Dec 21 2016 *)
%Y A227343 A000670 (columns 1 and 2), A048993, A059099 (row sums), A105794, A227342 (matrix inverse).
%K A227343 nonn,easy,tabl
%O A227343 0,4
%A A227343 _Peter Bala_, Jul 11 2013