A391520 Square array T(n,k) read by ascending antidiagonals: T(n,k) = (p - 1)/2*(2*(n - 1)*p^(n - 1) - (2*n - 3)*p^(n - 2)), n>=2, where p is the k-th odd prime.

5, 27, 18, 117, 170, 39, 459, 1250, 525, 105, 1701, 8250, 5439, 2255, 150, 6075, 51250, 50421, 36905, 3822, 264, 21141, 306250, 439383, 539055, 74022, 8840, 333, 72171, 1781250, 3680733, 7393705, 1278654, 224264, 12483, 495, 242757, 10156250, 30000495, 97435855, 20735286, 5070216, 354141, 22517, 798

Offset

2,1

Comments

T(n,k) is the total number of descents of all paths starting from a vertex on the 2n-th floor in the p-Bratteli diagram and ending at all the vertices on the first floor where p is the k-th odd prime.

Links

Table of n, a(n) for n=2..46.

Mantharam Parvathi, Annamalai Tamilselvi, and Devanbu Hepsi, p^(k)-Fibonacci Numbers of the p-Bratteli Diagram for Every Odd Prime p and Integer k>=0, arXiv:2603.04229 [math.CO], 2026. See pp. 1-2.

Wikipedia, Bratteli diagram

Formula

T(n,k) = (p - 1)/2*(2*(n - 1)*p^(n - 1) - (2*n - 3)*p^(n - 2)), n>=2, where p is the k-th odd prime.

G.f. for column p: x^2*(p-1)/2*((2p-1)-px)/(1-px)^2, |px|<1.

T(n,3) = 3*(2*(n-1)*7^(n-1)-(2*n-3)*7^(n-2)), n>=2.

G.f. for column p when p is the 3rd odd prime: 3*x^2*(13 - 7*x)/(1 - 7*x)^2. - Stefano Spezia, Dec 14 2025

Example

Array begins
v[n\p] [1]     [2]         [3]           [4]           [5]
   [2]  5,     18,         39,           105,           150,          . . .
   [3]  27,    170,        525,          2255,          3822,         . . .
   [4]  117,   1250,       5439,         36905,         74022,        . . .
   [5]  459,   8250,       50421,        539055,        1278654,      . . .
   [6]  1701,  51250,      439383,       7393705,       20735286,     . . .
   [7]  6075,  306250,     3680733,      97435855,      323024910,    . . .
   [8]  21141, 1781250,    30000495,     1248950505,    4894384326,   . . .
   [9]  72171, 10156250,   239651013,    15687172655,   72662782686,  . . .
   [.]    .      .            .             .              .

Maple

T:= (n, k)-> (p-> (p-1)/2*(2*(n-1)*p^(n-1)-(2*n-3)*p^(n-2)))(ithprime(k+1)):
seq(seq(T(1+d-k, k), k=1..d-1), d=2..10);  # Alois P. Heinz, Dec 30 2025

Crossrefs

Sequence in context: A097088 A196009 A091721 * A039283 A045162 A056735

Adjacent sequences: A391517 A391518 A391519 * A391521 A391522 A391523

Keyword

nonn,tabl,easy

Author

Tamilselvi Annamalai, Parvathi Mantharam, and Hepsi Devanbu, Dec 12 2025

Extensions

a(17) corrected by Sean A. Irvine, Dec 30 2025

Status

approved