A373168 Triangle read by rows: the exponential almost-Riordan array ( exp(x/(1-x)) | 1/(1-x), x ).

1, 1, 1, 3, 1, 1, 13, 2, 2, 1, 73, 6, 6, 3, 1, 501, 24, 24, 12, 4, 1, 4051, 120, 120, 60, 20, 5, 1, 37633, 720, 720, 360, 120, 30, 6, 1, 394353, 5040, 5040, 2520, 840, 210, 42, 7, 1, 4596553, 40320, 40320, 20160, 6720, 1680, 336, 56, 8, 1, 58941091, 362880, 362880, 181440, 60480, 15120, 3024, 504, 72, 9, 1

Offset

0,4

Links

Table of n, a(n) for n=0..65.

Y. Alp and E. G. Kocer, Exponential Almost-Riordan Arrays, Results Math 79, 173 (2024). See page 13.

Formula

T(n,0) = n! * [x^n] exp(x/(1-x)); T(n,k) = (n-1)!/(k-1)! * [x^(n-1)] 1/(1-x)*x^(k-1).

T(n,3) = A001710(n-1) for n > 2.

T(n,4) = A001715(n-1) for n > 3.

T(n,5) = A001720(n-1) for n > 4.

T(n,6) = A001725(n-1) for n > 5.

T(n,7) = A001730(n-1) for n > 6.

T(n,8) = A049388(n-8) for n > 7.

T(n,9) = A049389(n-9) for n > 8.

T(n,10) = A049398(n-10) for n > 9.

T(n,11) = A051431(n-11) for n > 10.

Example

The triangle begins:
     1;
     1,   1;
     3,   1,   1;
    13,   2,   2,  1;
    73,   6,   6,  3,  1;
   501,  24,  24, 12,  4, 1;
  4051, 120, 120, 60, 20, 5, 1;
  ...

Mathematica

T[n_, 0]:=n!SeriesCoefficient[Exp[x/(1-x)], {x, 0, n}]; T[n_, k_]:=(n-1)!/(k-1)!SeriesCoefficient[1/(1-x)*x^(k-1), {x, 0, n-1}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten

Crossrefs

Cf. A000142, A000262 (k=0), A001710, A001715, A001720, A001725, A001730, A049388, A051431.

Triangle A094587 with 1st column A000262.

Sequence in context: A129619 A094573 A055154 * A338875 A015112 A174690

Adjacent sequences: A373160 A373164 A373167 * A373169 A373170 A373171

Keyword

nonn,tabl

Author

Stefano Spezia, May 26 2024

Status

approved