A089918 (n+1)*a(n) equals the (n+1)-th term of the n-th binomial transform of this sequence.

1, 1, 4, 30, 344, 5470, 113892, 2988426, 96098256, 3706301286, 168494060060, 8900937960706, 539861090952504, 37212541117261614, 2889625585310057076, 250853703625289217690, 24183224498531542003616, 2573631282233504148553078, 300760619369532059346695820

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..292

Formula

From Seiichi Manyama, Mar 03 2026: (Start)

a(0) = 1; a(n) = Sum_{k=1..n} n^(k-1) * binomial(n,k) * a(n-k).

a(0) = 1; a(n) = n! * [x^n] (1/n) * exp(n*x) * Sum_{k=0..n-1} a(k)*x^k/k!. (End)

Example

Note the diagonal {(n+1)*a(n)} in array of iterated binomial transforms of this sequence:
  [1, 1,  4,  30,  344,  5470,  113892,  2988426, ...]
  [1,_2,  7,  46,  493,  7536,  152539,  3913722, ...]
  [1, 3,_12,  74,  728, 10542,  206188,  5158778, ...]
  [1, 4, 19,_120, 1109, 15058,  282039,  6851820, ...]
  [1, 5, 28, 190,_1720, 22014,  391732,  9185754, ...]
  [1, 6, 39, 290, 2669,_32820,  553867, 12457646, ...]
  [1, 7, 52, 426, 4088, 49486, _797244, 17128362, ...]
  [1, 8, 67, 604, 6133, 74742, 1164823,_23907408, ...]

Crossrefs

Cf. A000272, A000312, A293860.

Sequence in context: A209440 A052316 A379936 * A391472 A392789 A392784

Adjacent sequences: A089915 A089916 A089917 * A089919 A089920 A089921

Keyword

nonn

Author

Paul D. Hanna, Nov 14 2003

Extensions

More terms from Seiichi Manyama, Mar 03 2026

Status

approved