A339050 Triangle read by rows T(n, m) = F(2*m-1)*(n-m) + F(2*m), for 1 <= m <= n, where F = A000045 (Fibonacci).

1, 2, 3, 3, 5, 8, 4, 7, 13, 21, 5, 9, 18, 34, 55, 6, 11, 23, 47, 89, 144, 7, 13, 28, 60, 123, 233, 377, 8, 15, 33, 73, 157, 322, 610, 987, 9, 17, 38, 86, 191, 411, 843, 1597, 2584, 10, 19, 43, 99, 225, 500, 1076, 2207, 4181, 6765

Offset

1,2

Comments

This is the partial sum triangle of triangle A143929.

The main diagonal is the INVERT transform of the first column (offset 1 in both sequences).

Links

Table of n, a(n) for n=1..55.

Formula

T(n, m) = Sum_{k=1..m} A143929(n, k), n >=1, m = 1, 2, ..., n, otherwise 0.

T(n, m) = A(m)*n + B(m), with A(m) = A(m-1) + F(2*(m-1)), for m >= 2 and A(1) = 1, and B(m) = B(m-1) + (m-1)*F(2*(m-1)), for m >= 2 and B(1) = 0, where F(2*m) =A001906(m) and F(2*m-1) = A001519(m).

T(n, 1) = n, for n >= 1; T(n, m) = F(2*(m-1))*(n-m+1), if m >= 2 and n >= m, and 0 otherwise.

G.f. of column m: G(m,x) = x^m*(x*F(2*m-1)/(1-x)^2 + F(2*m)/(1-x)), for m >= 1.

G.f. of row polynomials R(n, x) := Sum{m=1..n} T(n, m)*x^m, that is g.f. of the triangle: G(z,x) = (x*z)*(1 - x*z^2)/((1- 3*x*z + (x*z)^2)*(1 - z)^2).

G.f. of (sub)diagonal k: D(k,x) = x*((k-1)*(1-x) + 1)/(1 - 3*x + x^2), for k >= 1.

Example

The triangle T(n, m) begins:
n\m   1  2  3  4   5   6    7    8    9   10 ...
1:    1
2:    2  3
3:    3  5  8
4:    4  7 13 21
5:    5  9 18 34  55
6:    6 11 23 47  89 144
7:    7 13 28 60 123 233  377
8:    8 15 33 73 157 322  610  987
9:    9 17 38 86 191 411  843 1597 2584
10:  10 19 43 99 225 500 1076 2207 4181 6765
...

Crossrefs

Cf. A000045, A001519, A001906, A143929.

The first columns (without leading zeros) are A001477(n), A005408(n+1), A005408(n+1), for n >= 1.

The first (sub)diagonals are A001906(m), A001519(m+1), A005248(m), for m >= 1.

Sequence in context: A153643 A053218 A198335 * A296335 A296635 A295051

Adjacent sequences: A339047 A339048 A339049 * A339051 A339052 A339053

Keyword

nonn,tabl,easy

Author

Gary W. Adamson and Wolfdieter Lang, Jan 15 2021

Status

approved