A203484 -id:A203484 - cp's OEIS frontend

A203498 Row sums of triangle A203484.

1, 2, 44, 12, 2316, 56, 86756, 7548, 2745504, 14216, 8303228, 27440, 26755926552, 17777912, 337554780656, 166149231952, 41032825702988, 10165065032, 19445273523788, 559223251080, 52853093762480872, 2217608618621076, 3130076262074284420, 9840817961344

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Jan 02 2012

nonn

Conjecture. If prime p==1 (mod 3), then a((p-1)/3)==2 (mod p); if prime p==2 mod 3, then a((2*p-1)/3)==2 (mod p).

Cf. A202941, A202917, A203269, A203278, A203484, A178473.

A178473 For n>=0, let n!^(4) = A202369(n+1) and, for 0<=m<=n, C^(4)(n,m) = n!^(4)/(m!^(4)*(n-m)!^(4)). The sequence gives triangle of numbers C^(4)(n,m) with rows of length n+1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 273, 273, 1, 1, 68, 9282, 68, 1, 1, 55, 1870, 1870, 55, 1, 1, 546, 15015, 3740, 15015, 546, 1, 1, 29, 7917, 1595, 1595, 7917, 29, 1

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Jan 02 2012

Conjecture. If p is prime of the form 4*k+1, then the k-th row contains two 1's and k-1 numbers multiple of p; if p is prime of the form 4*k+3, then the (2*k+1)-th row contains two 1's and 2*k numbers multiple of p.

			Triangle begins
n/m.|..0.....1.....2.....3.....4.....5.....6.....7
==================================================
.0..|..1
.1..|..1......1
.2..|..1......2......1
.3..|..1....273 ...273......1
.4..|..1.....68...9282.....68......1
.5..|..1.....55...1870...1870.....55......1
.6..|..1....546..15015...3740..15015....546....1
.7..|..1.....29...7917...1595...1595...7917...29.....1
.8..|

Cf. A175669, A053657, A202339, A202367, A202368, A202369, A202917, A202941, A203484.

Conjecture. A007814(C^(4)(n,m)) = A007814(C(n,m)).

A203509 Row sums of triangle A178473.

Original entry on oeis.org

1, 2, 4, 548, 9420, 3852, 34864, 19084, 15296, 154560176, 7180221844, 887102780, 211332046788, 71893259484, 20454788424, 5986072942766808, 5988933869570752, 22285488224, 2756032824242080, 17677170921656, 679436626785756, 20936052371230100988

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Jan 02 2012

nonn

Conjecture. If prime p==1 (mod 4), then a((p-1)/4)==2 (mod p); if prime p==3 (mod 4), then a((p-1)/2)==2 (mod p).

Cf. A202941, A202917, A203269, A203278, A203484, A178473, A203498.

cp's OEIS Frontend

A203498 Row sums of triangle A203484.

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A178473 For n>=0, let n!^(4) = A202369(n+1) and, for 0<=m<=n, C^(4)(n,m) = n!^(4)/(m!^(4)*(n-m)!^(4)). The sequence gives triangle of numbers C^(4)(n,m) with rows of length n+1.

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A203509 Row sums of triangle A178473.

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