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A176627 A q-form method for the symmetrical triangle sequence was found based on A000326 pentagonal numbers: q=3;c(n,q)=Product[(q*(3*q - 1)/2)^i, {i, 1, n}];t(n,m,q)=c(n,q)/(c(m,q)*c(n-m,q)- c(n,q)/(c(0,q)*c(n-0,q)+1 0
1, 1, 1, 1, 12, 1, 1, 144, 144, 1, 1, 1728, 20736, 1728, 1, 1, 20736, 2985984, 2985984, 20736, 1, 1, 248832, 429981696, 5159780352, 429981696, 248832, 1, 1, 2985984, 61917364224, 8916100448256, 8916100448256, 61917364224, 2985984, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Row sums are:

{1, 2, 14, 290, 24194, 6013442, 6020241410, 17956041596930,

215716134316769282, 7720769219294509793282, 1113047871457059085380747266,...}.

Integer sum:

Sum[3*n - 2, {n, 1, q}]=q*(3*q-1)/2

LINKS

Table of n, a(n) for n=0..36.

FORMULA

q=3;

c(n,q)=Product[(q*(3*q - 1)/2)^i, {i, 1, n}];

t(n,m,q)=c(n,q)/(c(m,q)*c(n-m,q)- c(n,q)/(c(0,q)*c(n-0,q)+1

EXAMPLE

{1},

{1, 1},

{1, 12, 1},

{1, 144, 144, 1},

{1, 1728, 20736, 1728, 1},

{1, 20736, 2985984, 2985984, 20736, 1},

{1, 248832, 429981696, 5159780352, 429981696, 248832, 1},

{1, 2985984, 61917364224, 8916100448256, 8916100448256, 61917364224, 2985984, 1},

{1, 35831808, 8916100448256, 15407021574586368, 184884258895036416, 15407021574586368, 8916100448256, 35831808, 1},

{1, 429981696, 1283918464548864, 26623333280885243904, 3833759992447475122176, 3833759992447475122176, 26623333280885243904, 1283918464548864, 429981696, 1},

{1, 5159780352, 184884258895036416, 46005119909369701466112, 79496847203390844133441536, 953962166440690129601298432, 79496847203390844133441536, 46005119909369701466112, 184884258895036416, 5159780352, 1}

MATHEMATICA

Clear[t, n, m, c, q];

c[n_, q_] = Product[(q*(3*q - 1)/2)^i, {i, 1, n}];

t[n_, m_, q_] = c[n, q]/(c[m, q]*c[n - m, q]) - c[n, q]/(c[0, q]*c[n - 0, q]) + 1;

Table[Flatten[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 2, 12}]

CROSSREFS

Cf. A000326, A118190

Sequence in context: A156280 A166962 A022175 * A015129 A172376 A289673

Adjacent sequences:  A176624 A176625 A176626 * A176628 A176629 A176630

KEYWORD

nonn,tabl,uned

AUTHOR

Roger L. Bagula, Apr 22 2010

STATUS

approved

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Last modified January 22 10:25 EST 2020. Contains 331144 sequences. (Running on oeis4.)