A134634 -id:A134634 - cp's OEIS frontend

A134635 Row sums of triangle A134634.

1, 2, 6, 18, 54, 164, 508, 1610, 5222, 17308, 58484, 200948, 700348, 2470472, 8804024, 31648858, 114623366, 417820972, 1531629764, 5642508508, 20878731476, 77561756152, 289156105544, 1081466311108, 4056621689564, 15257327887384, 57525469116168, 217383333920040, 823195469508792, 3123379468819600, 11872247508521072, 45203794091311354

Offset: 0

Gary W. Adamson, Nov 04 2007

nonn

			a(3) = 18 = sum of row 3 terms of triangle A134634: (5 + 4 + 4 + 5).

Noah Arbesfeld, Partial permutations avoiding pairs of patterns, Discrete Math., 313 (2013), 2614-2625.

Cf. A000108, A126966, A134634.

A134635 := n -> 2*binomial(2*n, n)/(n+1) + add(2^k*binomial(2*n-2*k, n-k)/(2*n-2*k-1), k=0..n): seq(A134635(n), n=0..31); # Mélika Tebni, Feb 11 2024

Conjecture: (n+1)*a(n) +(-7*n+1)*a(n-1) +2*(7*n-8)*a(n-2) +4*(-2*n+5)*a(n-3)=0. - R. J. Mathar, May 30 2014

a(n) = 2*A000108(n) - A126966(n). - Mélika Tebni, Feb 11 2024

Corrected and extended by N. J. A. Sloane, Feb 18 2013

A222403 Triangle read by rows: left and right edges are A000217, interior entries are filled in using the Pascal triangle rule.

Original entry on oeis.org

0, 1, 1, 3, 2, 3, 6, 5, 5, 6, 10, 11, 10, 11, 10, 15, 21, 21, 21, 21, 15, 21, 36, 42, 42, 42, 36, 21, 28, 57, 78, 84, 84, 78, 57, 28, 36, 85, 135, 162, 168, 162, 135, 85, 36, 45, 121, 220, 297, 330, 330, 297, 220, 121, 45, 55, 166, 341, 517, 627, 660, 627, 517, 341, 166, 55

Offset: 0

N. J. A. Sloane, Feb 18 2013

In general, if the sequence defining the left and right edges is [a_0, a_1, ...], the row sums [s_0, s_1, ...] are given by s_0=a_0 and, for n>0,
s_n = 2a_n + Sum_{i=1..n-1} 2^(n-i) a_i.
Conversely, given the rows sums [s_0, s_1, ...], the edge sequence is [a_0, a_1, ...] where a_0=s_0 and, for n>0, a_n = (s_n - Sum_{i=1..n-1} s_i)/2.

			Triangle begins:
0
1, 1
3, 2, 3
6, 5, 5, 6
10, 11, 10, 11, 10
15, 21, 21, 21, 21, 15
21, 36, 42, 42, 42, 36, 21
28, 57, 78, 84, 84, 78, 57, 28
...

Robert Israel, Table of n, a(n) for n = 0..10010

Other triangles of this type: A007318, A051666, A134634, A222404, A222405.
Cf. A000217.
Row sums are A005803.

d:=[seq(n*(n+1)/2,n=0..14)];
f:=proc(d) local T,M,n,i;
M:=nops(d);
T:=Array(0..M-1,0..M-1);
for n from 0 to M-1 do T[n,0]:=d[n+1]; T[n,n]:=d[n+1]; od:
for n from 2 to M-1 do
for i from 1 to n-1 do T[n,i]:=T[n-1,i-1]+T[n-1,i]; od: od:
lprint("triangle:");
for n from 0 to M-1 do lprint(seq(T[n,i],i=0..n)); od:
lprint("row sums:");
lprint([seq( add(T[i,j],j=0..i), i=0..M-1)]);
end;
f(d);

t[n_, n_] := n*(n+1)/2; t[n_, 0] := n*(n+1)/2; t[n_, k_] := t[n, k] = t[n-1, k-1] + t[n-1, k]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 20 2014 *)

G.f. as triangle: (1+x-4*x*y+x*y^2+x^2*y^2)*y/((1-y)^2*(-x*y+1)^2*(-x*y-y+1)). - Robert Israel, Apr 04 2018

cp's OEIS Frontend

A134635 Row sums of triangle A134634.

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