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').
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, 46, 13, 6, 1, 1, 1, 232, 331, 166, 73, 16, 7, 1, 1, 1, 764, 1303, 856, 281, 106, 19, 8, 1, 1, 1, 2620, 5937, 3844, 1741, 426, 145, 22, 9, 1, 1
Offset: 0
Examples
Array T(n, k) starts: [n\k] 0 1 2 3 4 5 6 7 -------------------------------------------------------------- [0] 1, 1, 1, 1, 1, 1, 1, 1, ... [A000012] [1] 1, 1, 1, 1, 1, 1, 1, 1, ... [A000012] [2] 1, 2, 3, 4, 5, 6, 7, 8, ... [A000027] [3] 1, 4, 7, 10, 13, 16, 19, 22, ... [A016777] [4] 1, 10, 25, 46, 73, 106, 145, 190, ... [A100536] [5] 1, 26, 81, 166, 281, 426, 601, 806, ... [6] 1, 76, 331, 856, 1741, 3076, 4951, 7456, ... [7] 1, 232, 1303, 3844, 8485, 15856, 26587, 41308, ... [8] 1, 764, 5937, 21820, 57233, 123516, 234529, 406652, ... [9] 1, 2620, 26785, 114076, 328753, 757756, 1510705, 2719900, ... [A000085][A047974][A115327][A115329][A115331]
References
- John Riordan, Introduction to Combinatorial Analysis, Dover (2002).
Links
- Urszula Bednarz and Małgorzata Wołowiec-Musiał, On a new generalization of telephone numbers, Turkish Journal of Mathematics: Vol. 43: No. 3, (2019).
- Carlos M. da Fonseca and Anthony G. Shannon, Telephone numbers extensions, J. Interdisc. Math. (2025) Vol. 28, No. 4, 1573-1580. See pp. 1574, 1577.
- Wikipedia, Telephone numbers.
Crossrefs
Programs
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Maple
T := (n, k) -> add(binomial(n, j)*doublefactorial(j-1)*k^(j/2), j = 0..n, 2): for n from 0 to 9 do lprint(seq(T(n, k), k = 0..7)) od; T := (n, k) -> ifelse(k=0, 1, I^(-n)*(2*k)^(n/2)*KummerU(-n/2, 1/2, -1/(2*k))): seq(seq(simplify(T(n-k, k)), k = 0..n), n = 0..10); T := proc(n, k) exp(x + (k/2)*x^2): series(%, x, 16): n!*coeff(%, x, n) end: seq(lprint(seq(simplify(T(n, k)), k = 0..8)), n = 0..9); T := proc(n, k) option remember; if n = 0 or n = 1 then 1 else T(n, k-1) + n*(k-1)*T(n, k-2) fi end: for n from 0 to 9 do seq(T(n, k), k=0..9) od; # Only to check the interpretation as a determinant of a lower Hessenberg matrix: gen := proc(i, j, n) local ev, tv; ev := irem(j+i, 2) = 0; tv := j < i and not ev; if j > i + 1 then 0 elif j = i + 1 then -1 elif j <= i and ev then 1 elif tv and i < n then x*(n + 1 - i) - 1 else x fi end: det := M -> LinearAlgebra:-Determinant(M): p := (n, k) -> subs(x = k, det(Matrix(n, (i, j) -> gen(i, j, n)))): for n from 0 to 9 do seq(p(n, k), k = 0..7) od;
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Mathematica
T[n_, k_] := Sum[Binomial[n, j] Factorial2[j-1] * If[j==0, 1, k^(j/2)], {j, 0, n, 2}]; Table[T[n-k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 25 2023 *) -
Python
from math import factorial, comb def oddfactorial(n: int) -> int: return factorial(2 * n) // (2**n * factorial(n)) def T(n: int, k: int) -> int: return sum(comb(n, 2 * j) * oddfactorial(j) * k**j for j in range(n + 1)) for n in range(10): print([T(n, k) for k in range(8)])
Formula
T(n, k) = Sum_{j=0..n, j even} binomial(n, j) * (j - 1)!! * k^(j/2).
T(n, k) = T(n, k-1) + n*(k-1)*T(n, k-2) for n >= 2, T(n, 0) = T(n, 1) = 1.
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.
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