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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

%I

%S 1,1,1,1,12,1,1,144,144,1,1,1728,20736,1728,1,1,20736,2985984,2985984,

%T 20736,1,1,248832,429981696,5159780352,429981696,248832,1,1,2985984,

%U 61917364224,8916100448256,8916100448256,61917364224,2985984,1,1

%N 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

%C Row sums are:

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

%C 215716134316769282, 7720769219294509793282, 1113047871457059085380747266,...}.

%C Integer sum:

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

%F q=3;

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

%F 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

%e {1},

%e {1, 1},

%e {1, 12, 1},

%e {1, 144, 144, 1},

%e {1, 1728, 20736, 1728, 1},

%e {1, 20736, 2985984, 2985984, 20736, 1},

%e {1, 248832, 429981696, 5159780352, 429981696, 248832, 1},

%e {1, 2985984, 61917364224, 8916100448256, 8916100448256, 61917364224, 2985984, 1},

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

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

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

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

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

%t 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;

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

%Y Cf. A000326, A118190

%K nonn,tabl,uned

%O 0,5

%A _Roger L. Bagula_, Apr 22 2010

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Last modified February 17 21:35 EST 2020. Contains 332006 sequences. (Running on oeis4.)