A055325 Matrix inverse of Euler's triangle A008292.

1, -1, 1, 3, -4, 1, -23, 33, -11, 1, 425, -620, 220, -26, 1, -18129, 26525, -9520, 1180, -57, 1, 1721419, -2519664, 905765, -113050, 5649, -120, 1, -353654167, 517670461, -186123259, 23248085, -1166221, 25347, -247, 1, 153923102577

Offset: 1

Author

Christian G. Bower, May 12 2000

Keywords

• sign
• tabl

Examples

			Triangle starts:
  [1]         1;
  [2]        -1,         1;
  [3]         3,        -4,          1;
  [4]       -23,        33,        -11,        1;
  [5]       425,      -620,        220,      -26,        1;
  [6]    -18129,     26525,      -9520,     1180,      -57,     1;
  [7]   1721419,  -2519664,     905765,  -113050,     5649,  -120,    1;
  [8]-353654167, 517670461, -186123259, 23248085, -1166221, 25347, -247, 1;
		

Links

• Robert Israel, Table of n, a(n) for n = 1..3321 (rows 1 to 81, flattened)

Crossrefs

Cf. A008292, A162498.

Programs

Maple

A008292:= proc(n, k) option remember;
  if k < 1 or k > n then 0
  elif k = 1 or k = n then 1
  else (k*procname(n-1, k)+(n-k+1)*procname(n-1, k-1))
  fi
end proc:
T:= Matrix(10,10,(i,j) -> A008292(i,j)):
R:= T^(-1):
seq(seq(R[i,j],j=1..i),i=1..10); # Robert Israel, May 25 2018

Mathematica

m = 10 (*rows*);
t[n_, k_] := Sum[(-1)^j*(k-j)^n*Binomial[n+1, j], {j, 0, k}];
M = Array[t, {m, m}] // Inverse;
Table[M[[i, j]], {i, 1, m}, {j, 1, i}] // Flatten (* Jean-François Alcover, Mar 05 2019 *)
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*)
iT[n_, n_]/;n>=1 := 1; iT[n_, k_]/;1<=kA055325*)
Flatten@Table[iT[n, k], {n, 1, 9}, {k, 1, n}] (* Oliver Seipel, Feb 10 2025 *)