A391520 - OEIS

%I #64 Mar 08 2026 10:28:14

%S 5,27,18,117,170,39,459,1250,525,105,1701,8250,5439,2255,150,6075,

%T 51250,50421,36905,3822,264,21141,306250,439383,539055,74022,8840,333,

%U 72171,1781250,3680733,7393705,1278654,224264,12483,495,242757,10156250,30000495,97435855,20735286,5070216,354141,22517,798

%N 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.

%C 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.

%H Mantharam Parvathi, Annamalai Tamilselvi, and Devanbu Hepsi, <a href="https://arxiv.org/abs/2603.04229">p^(k)-Fibonacci Numbers of the p-Bratteli Diagram for Every Odd Prime p and Integer k>=0</a>, arXiv:2603.04229 [math.CO], 2026. See pp. 1-2.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Bratteli_diagram">Bratteli diagram</a>

%F 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.

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

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

%F 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

%e Array begins

%e v[n\p] [1]     [2]         [3]           [4]           [5]

%e    [2]  5,     18,         39,           105,           150,          . . .

%e    [3]  27,    170,        525,          2255,          3822,         . . .

%e    [4]  117,   1250,       5439,         36905,         74022,        . . .

%e    [5]  459,   8250,       50421,        539055,        1278654,      . . .

%e    [6]  1701,  51250,      439383,       7393705,       20735286,     . . .

%e    [7]  6075,  306250,     3680733,      97435855,      323024910,    . . .

%e    [8]  21141, 1781250,    30000495,     1248950505,    4894384326,   . . .

%e    [9]  72171, 10156250,   239651013,    15687172655,   72662782686,  . . .

%e    [.]    .      .            .             .              .

%p T:= (n, k)-> (p-> (p-1)/2*(2*(n-1)*p^(n-1)-(2*n-3)*p^(n-2)))(ithprime(k+1)):

%p seq(seq(T(1+d-k,k), k=1..d-1), d=2..10);  # _Alois P. Heinz_, Dec 30 2025

%K nonn,tabl,easy

%O 2,1

%A _Tamilselvi Annamalai_, Parvathi Mantharam, and Hepsi Devanbu, Dec 12 2025

%E a(17) corrected by _Sean A. Irvine_, Dec 30 2025