A309495 Triangle read by rows, derived from A007318, row sums = the Bell Sequence.

1, 1, 1, 1, 2, 2, 1, 3, 5, 6, 1, 4, 9, 17, 21, 1, 5, 14, 34, 67, 82, 1, 6, 20, 58, 148, 290, 354, 1, 7, 27, 90, 275, 701, 1368, 1671, 1, 8, 35, 131, 460, 1411, 3579, 6986, 8536, 1, 9, 44, 182, 716, 2536, 7738, 19620, 38315, 46814, 1, 10, 54, 244, 1057, 4213, 14846, 45251, 114798, 224189, 273907

Offset: 1

Author

Gary W. Adamson, Aug 04 2019

Keywords

• nonn
• tabl

Comments

As described in A160185, we extract eigensequences of a rotated variant of Pascal's triangle:

1;

3, 1;

3, 2, 1;

1, 1, 1, 1;

Say, for these 4 columns, the eigensequence is (1, 4, 9, 15). Then preface the latter with a zero and take the first finite difference row, = (1, 3, 5, 6), fourth row of the triangle.

Examples

			Row 5 of A121207 is (1, 5, 14, 31, 52). Preface with a zero and take the first difference row:
     (0,  1,  5, 14, 31, 52)
  (..., 1,  4,  9, 17, 21) = row 5 of the triangle.
First few rows of the triangle:
  1;
  1, 1;
  1, 2,  2;
  1, 3,  5,  6;
  1, 4,  9, 17,  21;
  1, 5, 14, 34,  67,  82;
  1, 6, 20, 58, 148, 290, 354;
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Crossrefs

Row sums are A000110.

Main diagonal is A032346.

Cf. A007318, A121207, A160185.

Programs

PARI

\\ here U(n) is A121207.
U(n)={my(M=matrix(n,n)); for(n=1, n, M[n,1]=1; for(k=1, n-1, M[n,k+1]=sum(j=1, k, M[n-j, k-j+1]*binomial(n-2,j-1)))); M}
T(n)={my(A=U(n+1)); vector(n, n, my(t=A[n+1,2..n+1]); t-concat([0], t[1..n-1]))}
{ my(A=T(10)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Feb 20 2022

Formula

T(n,k) = A121207(n,k) - A121207(n, k-1) for k >= 2.

Extensions

Terms a(37) and beyond from Andrew Howroyd, Feb 20 2022