cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A359762 Array read by ascending antidiagonals. T(n, k) = n!*[x^n] exp(x + (k/2) * x^2). A generalization of the number of involutions (or of 'telephone numbers').

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%I A359762 #32 Jun 30 2025 18:41:13
%S A359762 1,1,1,1,1,1,1,2,1,1,1,4,3,1,1,1,10,7,4,1,1,1,26,25,10,5,1,1,1,76,81,
%T A359762 46,13,6,1,1,1,232,331,166,73,16,7,1,1,1,764,1303,856,281,106,19,8,1,
%U A359762 1,1,2620,5937,3844,1741,426,145,22,9,1,1
%N A359762 Array read by ascending antidiagonals. T(n, k) = n!*[x^n] exp(x + (k/2) * x^2). A generalization of the number of involutions (or of 'telephone numbers').
%C A359762 The array is a generalization of the number of involutions of permutations on n letters, A000085, also known as 'telephone numbers'. According to Bednarz et al. the telephone number interpretation "is due to John Riordan, who noticed that T(n, 1) is the number of connection patterns in a telephone system with n subscribers."
%C A359762 In graph theory, the n-th telephone number is the total number of matchings of a complete graph K_n (see the Wikipedia entry). Assuming a network with k possibilities of connections leads to a network that can be modeled by a complete multigraph K(n, k). The total number of connection patterns in such a network is given by T(n, k).
%D A359762 John Riordan, Introduction to Combinatorial Analysis, Dover (2002).
%H A359762 Urszula Bednarz and Małgorzata Wołowiec-Musiał, <a href="https://doi.org/10.3906/mat-1812-108">On a new generalization of telephone numbers</a>, Turkish Journal of Mathematics: Vol. 43: No. 3, (2019).
%H A359762 Carlos M. da Fonseca and Anthony G. Shannon, <a href="https://doi.org/10.47974/JIM-2148">Telephone numbers extensions</a>, J. Interdisc. Math. (2025) Vol. 28, No. 4, 1573-1580. See pp. 1574, 1577.
%H A359762 Wikipedia, <a href="https://en.wikipedia.org/wiki/Telephone_number_(mathematics)">Telephone numbers</a>.
%F A359762 T(n, k) = Sum_{j=0..n, j even} binomial(n, j) * (j - 1)!! * k^(j/2).
%F A359762 T(n, k) = T(n, k-1) + n*(k-1)*T(n, k-2) for n >= 2, T(n, 0) = T(n, 1) = 1.
%F A359762 T(n, k) = i^(-n) * (2*k)^(n/2) * KummerU(-n/2, 1/2, -1/(2*k)) for k >= 1, and T(n, 0) = 1.
%e A359762 Array T(n, k) starts:
%e A359762   [n\k] 0   1      2        3       4        5        6        7
%e A359762   --------------------------------------------------------------
%e A359762   [0] 1,    1,     1,       1,      1,       1,       1,       1, ... [A000012]
%e A359762   [1] 1,    1,     1,       1,      1,       1,       1,       1, ... [A000012]
%e A359762   [2] 1,    2,     3,       4,      5,       6,       7,       8, ... [A000027]
%e A359762   [3] 1,    4,     7,      10,     13,      16,      19,      22, ... [A016777]
%e A359762   [4] 1,   10,    25,      46,     73,     106,     145,     190, ... [A100536]
%e A359762   [5] 1,   26,    81,     166,    281,     426,     601,     806, ...
%e A359762   [6] 1,   76,   331,     856,   1741,    3076,    4951,    7456, ...
%e A359762   [7] 1,  232,  1303,    3844,   8485,   15856,   26587,   41308, ...
%e A359762   [8] 1,  764,  5937,   21820,  57233,  123516,  234529,  406652, ...
%e A359762   [9] 1, 2620, 26785,  114076, 328753,  757756, 1510705, 2719900, ...
%e A359762    [A000085][A047974][A115327][A115329][A115331]
%p A359762 T := (n, k) -> add(binomial(n, j)*doublefactorial(j-1)*k^(j/2), j = 0..n, 2):
%p A359762 for n from 0 to 9 do lprint(seq(T(n, k), k = 0..7)) od;
%p A359762 T := (n, k) -> ifelse(k=0, 1, I^(-n)*(2*k)^(n/2)*KummerU(-n/2, 1/2, -1/(2*k))):
%p A359762 seq(seq(simplify(T(n-k, k)), k = 0..n), n = 0..10);
%p A359762 T := proc(n, k) exp(x + (k/2)*x^2): series(%, x, 16): n!*coeff(%, x, n) end:
%p A359762 seq(lprint(seq(simplify(T(n, k)), k = 0..8)), n = 0..9);
%p A359762 T := proc(n, k) option remember; if n = 0 or n = 1 then 1 else T(n, k-1) +
%p A359762 n*(k-1)*T(n, k-2) fi end: for n from 0 to 9 do seq(T(n, k), k=0..9) od;
%p A359762 # Only to check the interpretation as a determinant of a lower Hessenberg matrix:
%p A359762 gen := proc(i, j, n) local ev, tv; ev := irem(j+i, 2) = 0; tv := j < i and not ev;
%p A359762 if j > i + 1 then 0 elif j = i + 1 then -1 elif j <= i and ev then 1
%p A359762 elif tv and i < n then x*(n + 1 - i) - 1 else x fi end:
%p A359762 det := M -> LinearAlgebra:-Determinant(M):
%p A359762 p := (n, k) -> subs(x = k, det(Matrix(n, (i, j) -> gen(i, j, n)))):
%p A359762 for n from 0 to 9 do seq(p(n, k), k = 0..7) od;
%t A359762 T[n_, k_] := Sum[Binomial[n, j] Factorial2[j-1] * If[j==0, 1,  k^(j/2)], {j, 0, n, 2}];
%t A359762 Table[T[n-k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jan 25 2023 *)
%o A359762 (Python)
%o A359762 from math import factorial, comb
%o A359762 def oddfactorial(n: int) -> int:
%o A359762     return factorial(2 * n) // (2**n * factorial(n))
%o A359762 def T(n: int, k: int) -> int:
%o A359762     return sum(comb(n, 2 * j) * oddfactorial(j) * k**j for j in range(n + 1))
%o A359762 for n in range(10): print([T(n, k) for k in range(8)])
%Y A359762 Rows include: A000012, A000027, A016777, A100536.
%Y A359762 Columns include: A000012, A000085, A047974, A115327, A115329, A115331, A293720.
%Y A359762 Cf. A000085, A277614, A359760.
%K A359762 nonn,tabl
%O A359762 0,8
%A A359762 _Peter Luschny_, Jan 14 2023