cp's OEIS Frontend

A137682 Left border of triangle A137680.

%I A137682 #14 Sep 19 2024 07:22:24
%S A137682 1,1,3,7,17,40,96,228,544,1296,3089,7361,17544,41810,99643,237471,
%T A137682 565946,1348773,3214424,7660679,18257085,43510652,103695461,247129108,
%U A137682 588963062,1403628615,3345155947,7972242937,18999609718,45280252031
%N A137682 Left border of triangle A137680.
%C A137682 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 A137682 Starting (1, 3, 7, ...) = INVERT transform of A160096. - _Gary W. Adamson_, May 01 2009
%F A137682 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 A137682 First few rows of triangle A137680 =
%e A137682   1;
%e A137682   1, 1;
%e A137682   3, 0, 1;
%e A137682   7, 2, 0, 1;
%e A137682   ...
%e A137682 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 A137682 := proc(n)
%p A137682     A137680(n,1) ;
%p A137682 end proc:
%p A137682 seq(A137682(n),n=1..30) ; # _R. J. Mathar_, Aug 12 2012
%t A137682 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 A137682 a[n_] := T[n, 1];
%t A137682 Table[a[n], {n, 1, 30}] (* _Jean-François Alcover_, Sep 19 2024, after _R. J. Mathar_ in A137680 *)
%Y A137682 Cf. A137680, A137681.
%Y A137682 Cf. A160096.
%K A137682 nonn
%O A137682 1,3
%A A137682 _Gary W. Adamson_, Feb 05 2008