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A222403 Triangle read by rows: left and right edges are A000217, interior entries are filled in using the Pascal triangle rule. 5
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

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.

LINKS

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

FORMULA

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

EXAMPLE

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

...

MAPLE

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);

MATHEMATICA

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-Fran├žois Alcover, Jan 20 2014 *)

CROSSREFS

Other triangles of this type: A007318, A051666, A134634, A222404, A222405.

Cf. A000217.

Row sums are A005803.

Sequence in context: A131990 A033771 A033795 * A033783 A033807 A058691

Adjacent sequences:  A222400 A222401 A222402 * A222404 A222405 A222406

KEYWORD

nonn,tabl

AUTHOR

N. J. A. Sloane, Feb 18 2013

STATUS

approved

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Last modified November 20 02:34 EST 2019. Contains 329323 sequences. (Running on oeis4.)