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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).

%I A339050 #6 Jan 22 2021 22:46:05
%S A339050 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,
%T A339050 233,377,8,15,33,73,157,322,610,987,9,17,38,86,191,411,843,1597,2584,
%U A339050 10,19,43,99,225,500,1076,2207,4181,6765
%N 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).
%C A339050 This is the partial sum triangle of triangle A143929.
%C A339050 The main diagonal is the INVERT transform of the first column (offset 1 in both sequences).
%F A339050 T(n, m) = Sum_{k=1..m} A143929(n, k), n >=1, m = 1, 2, ..., n, otherwise 0.
%F A339050 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).
%F A339050 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.
%F A339050 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.
%F A339050 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).
%F A339050 G.f. of (sub)diagonal k: D(k,x) = x*((k-1)*(1-x) + 1)/(1 - 3*x + x^2), for k >= 1.
%e A339050 The triangle T(n, m) begins:
%e A339050 n\m   1  2  3  4   5   6    7    8    9   10 ...
%e A339050 1:    1
%e A339050 2:    2  3
%e A339050 3:    3  5  8
%e A339050 4:    4  7 13 21
%e A339050 5:    5  9 18 34  55
%e A339050 6:    6 11 23 47  89 144
%e A339050 7:    7 13 28 60 123 233  377
%e A339050 8:    8 15 33 73 157 322  610  987
%e A339050 9:    9 17 38 86 191 411  843 1597 2584
%e A339050 10:  10 19 43 99 225 500 1076 2207 4181 6765
%e A339050 ...
%Y A339050 Cf. A000045, A001519, A001906, A143929.
%Y A339050 The first columns (without leading zeros) are A001477(n), A005408(n+1), A005408(n+1), for n >= 1.
%Y A339050 The first (sub)diagonals are A001906(m), A001519(m+1), A005248(m), for m >= 1.
%K A339050 nonn,tabl,easy
%O A339050 1,2
%A A339050 _Gary W. Adamson_ and _Wolfdieter Lang_, Jan 15 2021