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Left border of triangle A137680.
5

%I #14 Sep 19 2024 07:22:24

%S 1,1,3,7,17,40,96,228,544,1296,3089,7361,17544,41810,99643,237471,

%T 565946,1348773,3214424,7660679,18257085,43510652,103695461,247129108,

%U 588963062,1403628615,3345155947,7972242937,18999609718,45280252031

%N Left border of triangle A137680.

%C Each term in the sequence (n > 1) = sum of previous terms of triangle A137680 = partial sums of sequence A137681: (1, 2, 4, 10, 23, ...).

%C Starting (1, 3, 7, ...) = INVERT transform of A160096. - _Gary W. Adamson_, May 01 2009

%F Partial sums of sequence A137681 prefaced with a 1. a(n) is the sum of all terms in rows 1 through (n-1) in triangle A137680.

%e First few rows of triangle A137680 =

%e 1;

%e 1, 1;

%e 3, 0, 1;

%e 7, 2, 0, 1;

%e ...

%e a(5) = 17 is the sum of 1 through 4 row terms of triangle A137680: (1 + 2 + 4 + 10); where (1, 2, 4, 10, 23, ...) = A137681 = row sums of triangle A137680 = first difference row of A137682, n > 1.

%p A137682 := proc(n)

%p A137680(n,1) ;

%p end proc:

%p seq(A137682(n),n=1..30) ; # _R. J. Mathar_, Aug 12 2012

%t T[n_, k_] := T[n, k] = Which[k < 1 || k > n, 0, n == 1, 1, k == 1, Sum[T[r, j], {r, 1, n-1}, {j, 1, r}], True, T[n-1, k-1] - T[n-k, k-1]];

%t a[n_] := T[n, 1];

%t Table[a[n], {n, 1, 30}] (* _Jean-François Alcover_, Sep 19 2024, after _R. J. Mathar_ in A137680 *)

%Y Cf. A137680, A137681.

%Y Cf. A160096.

%K nonn

%O 1,3

%A _Gary W. Adamson_, Feb 05 2008