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.
%I A359760 #21 Apr 13 2025 03:33:15
%S A359760 1,1,0,1,0,1,1,0,3,0,1,0,6,0,3,1,0,10,0,15,0,1,0,15,0,45,0,15,1,0,21,
%T A359760 0,105,0,105,0,1,0,28,0,210,0,420,0,105,1,0,36,0,378,0,1260,0,945,0,1,
%U A359760 0,45,0,630,0,3150,0,4725,0,945,1,0,55,0,990,0,6930,0,17325,0,10395,0
%N A359760 Triangle read by rows. The Kummer triangle, the coefficients of the Kummer polynomials. K(n, k) = binomial(n, k) * oddfactorial(k/2) if k is even, otherwise 0, where oddfactorial(z) := (2*z)!/(2^z*z!).
%C A359760 The Kummer numbers K(n, k) are a refinement of the oddfactorial numbers (A001147) in the sense that they are the coefficients of polynomials K(n, x) = Sum_{n..k} K(n, k) * x^k that take the value oddfactorial(n) at x = 1. The coefficients of x^n are the aerated oddfactorial numbers A123023.
%C A359760 These numbers appear in many different versions (see the crossrefs). They are the coefficients of the Chebyshev-Hermite polynomials in signed form when ordered in decreasing powers. Our exposition is based on the seminal paper by Kummer, which preceded the work of Chebyshev and Hermite for more than 20 years. They are also referred to as Bessel numbers of the second kind (Mansour et al.) when the odd powers are omitted.
%D A359760 John Riordan, Introduction to Combinatorial Analysis, Dover (2002), pp. 85-86.
%H A359760 Pierre Humbert, <a href="http://www.numdam.org/item/?id=NAM_1922_5_1__81_0">Monographie des polynômes de Kummer</a>, Nouvelles annales de mathématiques, journal des candidats aux écoles polytechnique et normale, Serie 5, Volume 1 (1922), pp. 81-92.
%H A359760 E. E. Kummer, <a href="https://gdz.sub.uni-goettingen.de/id/PPN243919689_0015?tify=%7B%22view%22:%22info%22,%22pages%22:%5B43%5D%7D">Über die hypergeometrische Reihe</a>, Journal für die reine und angewandte Mathematik 15 (1836): 39-83.
%H A359760 T. Mansour, M. Schork and M. Shattuck, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Schork/schork2.html">The Generalized Stirling and Bell Numbers Revisited</a>, Journal of Integer Sequences, Vol. 15 (2012), #12.8.3.
%H A359760 Ladislav Truksa, <a href="http://dml.cz/dmlcz/144552">Hypergeometric orthogonal systems of polynomials III</a>, Aktuárské vědy, Vol. 2 (1931), No. 4, 177-203, (see p.200).
%F A359760 Let p(n, x) = 2^(n/2)*(-1/x^2)^(-n/2)*KummerU(-n/2, 1/2, -1/(2*x^2)).
%F A359760 p(n, 1) = A000085(n); p(n, sqrt(2)) = A047974(n); p(n, 2) = A115329(n);
%F A359760 p(2, n) = A002522(n) (n >= 1); p(3, n) = A056107(n) (n >= 1);
%F A359760 p(n, n) = A359739(n) (n >= 1); 2^n*p(n, 1/2) = A005425(n).
%F A359760 K(n, k) = [x^k] p(n, x).
%F A359760 K(n, k) = [t^k] (n! * [x^n] exp(x + (t*x)^2 / 2)).
%F A359760 K(n, n) = A123023(n).
%F A359760 K(n, n-1) = A123023(n + 1).
%F A359760 K(2*n, 2*n) = A001147(n).
%F A359760 K(4*n, 2*n) = A359761, the central terms without zeros.
%F A359760 K(2*n+2, 2*n) = A001879.
%F A359760 Sum_{k=0..n} (-1)^n * i^k * K(n, k) = A001464(n), ((the number of even involutions) - (the number of odd involutions) in the symmetric group S_n (_Robert Israel_)).
%F A359760 Sum_{k=0..n} Sum_{j=0..k} K(n, j) = A000085(n + 1).
%F A359760 For a recursion see the Python program.
%e A359760 Triangle K(n, k) starts:
%e A359760 [0] 1;
%e A359760 [1] 1, 0;
%e A359760 [2] 1, 0, 1;
%e A359760 [3] 1, 0, 3, 0;
%e A359760 [4] 1, 0, 6, 0, 3;
%e A359760 [5] 1, 0, 10, 0, 15, 0;
%e A359760 [6] 1, 0, 15, 0, 45, 0, 15;
%e A359760 [7] 1, 0, 21, 0, 105, 0, 105, 0;
%e A359760 [8] 1, 0, 28, 0, 210, 0, 420, 0, 105;
%e A359760 [9] 1, 0, 36, 0, 378, 0, 1260, 0, 945, 0;
%p A359760 oddfactorial := proc(z) (2*z)! / (2^z*z!) end:
%p A359760 K := (n, k) -> ifelse(irem(k, 2) = 1, 0, binomial(n, k) * oddfactorial(k/2)):
%p A359760 seq(seq(K(n, k), k = 0..n), n = 0..11);
%p A359760 # Alternative, as coefficients of polynomials:
%p A359760 p := (n, x) -> 2^(n/2)*(-1/x^2)^(-n/2)*KummerU(-n/2, 1/2, -1/(2*x^2)):
%p A359760 seq(print(seq(coeff(simplify(p(n, x)), x, k), k = 0..n)), n = 0 ..9);
%p A359760 # Using the exponential generating function:
%p A359760 egf := exp(x + (t*x)^2 / 2): ser := series(egf, x, 12):
%p A359760 seq(print(seq(coeff(n! * coeff(ser, x, n), t, k), k = 0..n)), n = 0..9);
%t A359760 K[n_, k_] := K[n, k] = Which[OddQ[k], 0, k == 0, 1, n == k, K[n - 1, n - 2], True, K[n - 1, k] n/(n - k)];
%t A359760 Table[K[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jan 25 2023 *)
%o A359760 (Python)
%o A359760 from functools import cache
%o A359760 @cache
%o A359760 def K(n: int, k: int) -> int:
%o A359760 if k % 2: return 0
%o A359760 if n < 3: return 1
%o A359760 if n == k: return K(n - 1, n - 2)
%o A359760 return (K(n - 1, k) * n) // (n - k)
%o A359760 for n in range(10): print([K(n, k) for k in range(n + 1)])
%Y A359760 Variants: Signed version: A073278. Other variants are the irregular triangle A100861 with zeros deleted, A066325 and A099174 with reversed rows, A111924, A144299, A104556.
%Y A359760 Cf. A000085, A001147, A123023, A047974, A115329, A002522, A056107, A359739,
%Y A359760 A001879, A001464, A005425, A344501, A123022, A359761.
%K A359760 nonn,tabl
%O A359760 0,9
%A A359760 _Peter Luschny_, Jan 13 2023