A140676 -id:A140676 - cp's OEIS frontend

A246799 Triangle read by rows: T(n,k) is the coefficient A_k in the transformation Sum_{k=0..n} (k+1)x^k = Sum_{k=0..n} A_k(x-3)^k.

1, 7, 2, 34, 20, 3, 142, 128, 39, 4, 547, 668, 309, 64, 5, 2005, 3098, 1929, 604, 95, 6, 7108, 13304, 10434, 4384, 1040, 132, 7, 24604, 54128, 51258, 27064, 8600, 1644, 175, 8, 83653, 211592, 234966, 149536, 59630, 15252, 2443, 224, 9, 280483, 802082, 1022286, 761896, 365810, 117312, 25123, 3464, 279, 10

Offset: 0

Derek Orr, Nov 15 2014

Consider the transformation 1 + 2x + 3x^2 + 4x^3 + ... + (n+1)*x^n = A_0*(x-3)^0 + A_1*(x-3)^1 + A_2*(x-3)^2 + ... + A_n*(x-3)^n. This sequence gives A_0, ... A_n as the entries in the n-th row of this triangle, starting at n = 0.

			Triangle starts:
1;
7,           2;
34,         20,       3;
142,       128,      39,      4;
547,       668,     309,     64,      5;
2005,     3098,    1929,    604,     95,      6;
7108,    13304,   10434,   4384,   1040,    132,     7;
24604,   54128,   51258,  27064,   8600,   1644,   175,    8;
83653,  211592,  234966, 149536,  59630,  15252,  2443,  224,   9;
280483, 802082, 1022286, 761896, 365810, 117312, 25123, 3464, 279, 10;
...

Cf. A246797, A014915, A140676, A246788.

T(n, k) = (k+1)*sum(i=0, n-k, 3^i*binomial(i+k+1, k+1))
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")))

T(n,0) = ((2*n+1)*3^(n+1) + 1)/4, for n >= 0.
T(n,n-1) = n*(3*n+4), for n >= 1.
Row n sums to A014916(n+1) = T(2*n+1,0) of A246788.

A192491 Molecular topological indices of the complete tripartite graphs K_{n,n,n}.

Original entry on oeis.org

24, 240, 864, 2112, 4200, 7344, 11760, 17664, 25272, 34800, 46464, 60480, 77064, 96432, 118800, 144384, 173400, 206064, 242592, 283200, 328104, 377520, 431664, 490752, 555000, 624624, 699840, 780864, 867912, 961200

Offset: 1

Eric W. Weisstein, Jul 10 2011

Eric Weisstein's World of Mathematics, Complete Tripartite Graph
Eric Weisstein's World of Mathematics, Molecular Topological Index
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000326, A000330, A000292, A017233, A008594, A140676, A046092.

a(n) = 12*n^2*(3*n-1).
a(n) = 24*A050509(n).
G.f.: 24*x*(2*x^2+6*x+1)/(x-1)^4. [Colin Barker, Nov 04 2012]
From Bruce J. Nicholson, Sep 18 2019: (Start)
a(n) = 24*n * A000326(n).
a(n) = 4*n^2 * A017233(n).
a(n) = 24*(n^3 + A000292(n-2) + A000330(n-2)).
a(n) = 24*(n^4 - (A008585(n) * A000330(n-1))).
a(n) = 6*A046092(n) + (A008594(n+1) * A140676(n-1)). (End)

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A246799 Triangle read by rows: T(n,k) is the coefficient A_k in the transformation Sum_{k=0..n} (k+1)x^k = Sum_{k=0..n} A_k(x-3)^k.

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A192491 Molecular topological indices of the complete tripartite graphs K_{n,n,n}.

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cp's OEIS Frontend

A246799 Triangle read by rows: T(n,k) is the coefficient A_k in the transformation Sum_{k=0..n} (k+1)*x^k = Sum_{k=0..n} A_k*(x-3)^k.

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Author

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Comments

Examples

Crossrefs

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PARI

Formula

A192491 Molecular topological indices of the complete tripartite graphs K_{n,n,n}.

Views

Author

Keywords

Links

Crossrefs

Formula

A246799 Triangle read by rows: T(n,k) is the coefficient A_k in the transformation Sum_{k=0..n} (k+1)x^k = Sum_{k=0..n} A_k(x-3)^k.