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A055325 Matrix inverse of Euler's triangle A008292.

%I A055325 #18 Feb 23 2025 11:34:32
%S A055325 1,-1,1,3,-4,1,-23,33,-11,1,425,-620,220,-26,1,-18129,26525,-9520,
%T A055325 1180,-57,1,1721419,-2519664,905765,-113050,5649,-120,1,-353654167,
%U A055325 517670461,-186123259,23248085,-1166221,25347,-247,1,153923102577
%N A055325 Matrix inverse of Euler's triangle A008292.
%H A055325 Robert Israel, <a href="/A055325/b055325.txt">Table of n, a(n) for n = 1..3321</a> (rows 1 to 81, flattened)
%e A055325 Triangle starts:
%e A055325   [1]         1;
%e A055325   [2]        -1,         1;
%e A055325   [3]         3,        -4,          1;
%e A055325   [4]       -23,        33,        -11,        1;
%e A055325   [5]       425,      -620,        220,      -26,        1;
%e A055325   [6]    -18129,     26525,      -9520,     1180,      -57,     1;
%e A055325   [7]   1721419,  -2519664,     905765,  -113050,     5649,  -120,    1;
%e A055325   [8]-353654167, 517670461, -186123259, 23248085, -1166221, 25347, -247, 1;
%p A055325 A008292:= proc(n, k) option remember;
%p A055325   if k < 1 or k > n then 0
%p A055325   elif k = 1 or k = n then 1
%p A055325   else (k*procname(n-1, k)+(n-k+1)*procname(n-1, k-1))
%p A055325   fi
%p A055325 end proc:
%p A055325 T:= Matrix(10,10,(i,j) -> A008292(i,j)):
%p A055325 R:= T^(-1):
%p A055325 seq(seq(R[i,j],j=1..i),i=1..10); # _Robert Israel_, May 25 2018
%t A055325 m = 10 (*rows*);
%t A055325 t[n_, k_] := Sum[(-1)^j*(k-j)^n*Binomial[n+1, j], {j, 0, k}];
%t A055325 M = Array[t, {m, m}] // Inverse;
%t A055325 Table[M[[i, j]], {i, 1, m}, {j, 1, i}] // Flatten (* _Jean-François Alcover_, Mar 05 2019 *)
%t A055325 T[1, 1] := 1; T[n_, k_]/;1<=k<=n := T[n, k] = (n-k+1) T[n-1, k-1] + k T[n-1, k]; T[n_, k_] := 0;(*A008292*)
%t A055325 iT[n_, n_]/;n>=1 := 1; iT[n_, k_]/;1<=k<n := iT[n, k] = -T[n, k] - Sum[T[n, k+j] iT[k+j, k], {j, n-k-1}]; iT[n_, k_] := 0;(*A055325*)
%t A055325 Flatten@Table[iT[n, k], {n, 1, 9}, {k, 1, n}] (* _Oliver Seipel_, Feb 10 2025 *)
%Y A055325 Cf. A008292, A162498.
%K A055325 sign,tabl
%O A055325 1,4
%A A055325 _Christian G. Bower_, May 12 2000