A113899 Number parallelogram based on Pascal's triangle.

252, 126, 126, 56, 140, 56, 21, 105, 105, 21, 6, 60, 120, 60, 6, 1, 25, 100, 100, 25, 1, 6, 60, 120, 60, 6, 21, 105, 105, 21, 56, 140, 56, 126, 126, 252

Offset

0,1

Comments

This sequence can be considered as a parallelogram of the following products of binomial coefficients:

.............................C(0,0)*C(10,5)

......................C(1,0)*C(9,5)...C(1,1)*C(9,4)

...............C(2,0)*C(8,5)...C(2,1)*C(8,4)...C(2,2)*C(8,3)

........C(3,0)*C(7,5)...C(3,1)*C(7,4)...C(3,2)*C(7,3)...C(3,3)*C(7,2)

...C(4,0)*C(6,5)...C(4,1)*C(6,4)...C(4,2)*C(6,3)...C(4,3)*C(8,2)...C(4,4)*C(6,1)

C(5,0)*C(5,5)...C(5,1)*C(5,4)...C(5,2)*C(5,3)...C(5,3)*C(5,2)...C(5,4)*C(5,1)...C(5,5)*C(5,0)

...C(6,1)*C(4,4)...C(4,1)*C(6,4)...C(4,2)*C(6,3)...C(4,3)*C(8,2)...C(6,5)*C(4,0)

........C(7,2)*C(3,3)...C(7,3)*C(3,2)...C(7,4)*C(3,1)...C(7,5)*C(3,0)

...............C(8,3)*C(2,2)...C(8,4)*C(2,1)...C(8,5)*C(2,0)

......................C(9,4)*C(1,1)...C(9,5)*C(1,0)

.............................C(10,5)*C(0,0)

.

which gives:

.

10..0.................................252..............................

.9..1...........................126.........126........................

.8..2......................56.........140..........56..................

.7..3................21.........105.........105..........21............

.6..4..........6...........60.........120..........60..........6.......

.5..5....1...........25.........100.........100..........25...........1

.4..6..........6...........60.........120..........60..........6.......

.3..7................21.........105.........105..........21............

.2..8......................56.........140..........56..................

.1..9...........................126.........126........................

0..10..................................252.............................

matching.0.....1.....2.....3.....4.....5.....6.....7.....8.....9.....10

Links

Table of n, a(n) for n=0..35.

Maple

for n from 0 to 5 do seq(binomial(i, n)*binomial(10-i, 5-n), i=n..5+n ); od; # Zerinvary Lajos, Mar 31 2009

Crossrefs

Cf. A113162, A113163, A113164.

Sequence in context: A177809 A367577 A365915 * A045182 A394546 A330616

Adjacent sequences: A113896 A113897 A113898 * A113900 A113901 A113902

Keyword

easy,fini,full,nonn,less

Author

Zerinvary Lajos, Jan 29 2006, May 28 2007

Extensions

Edited by Sean A. Irvine, Mar 26 2026

Status

approved