A099174 -id:A099174 - cp's OEIS frontend

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) := (2z)!/(2^zz!).

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, 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, 0, 45, 0, 630, 0, 3150, 0, 4725, 0, 945, 1, 0, 55, 0, 990, 0, 6930, 0, 17325, 0, 10395, 0

Offset: 0

Peter Luschny, Jan 13 2023

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.

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.

			Triangle K(n, k) starts:
 [0] 1;
 [1] 1, 0;
 [2] 1, 0,  1;
 [3] 1, 0,  3, 0;
 [4] 1, 0,  6, 0,   3;
 [5] 1, 0, 10, 0,  15, 0;
 [6] 1, 0, 15, 0,  45, 0,   15;
 [7] 1, 0, 21, 0, 105, 0,  105, 0;
 [8] 1, 0, 28, 0, 210, 0,  420, 0, 105;
 [9] 1, 0, 36, 0, 378, 0, 1260, 0, 945, 0;

John Riordan, Introduction to Combinatorial Analysis, Dover (2002), pp. 85-86.

Pierre Humbert, Monographie des polynômes de Kummer, Nouvelles annales de mathématiques, journal des candidats aux écoles polytechnique et normale, Serie 5, Volume 1 (1922), pp. 81-92.
E. E. Kummer, Über die hypergeometrische Reihe, Journal für die reine und angewandte Mathematik 15 (1836): 39-83.
T. Mansour, M. Schork and M. Shattuck, The Generalized Stirling and Bell Numbers Revisited, Journal of Integer Sequences, Vol. 15 (2012), #12.8.3.
Ladislav Truksa, Hypergeometric orthogonal systems of polynomials III, Aktuárské vědy, Vol. 2 (1931), No. 4, 177-203, (see p.200).

Variants: Signed version: A073278. Other variants are the irregular triangle A100861 with zeros deleted, A066325 and A099174 with reversed rows, A111924, A144299, A104556.
Cf. A000085, A001147, A123023, A047974, A115329, A002522, A056107, A359739,
A001879, A001464, A005425, A344501, A123022, A359761.

oddfactorial := proc(z) (2*z)! / (2^z*z!) end:
K := (n, k) -> ifelse(irem(k, 2) = 1, 0, binomial(n, k) * oddfactorial(k/2)):
seq(seq(K(n, k), k = 0..n), n = 0..11);
# Alternative, as coefficients of polynomials:
p := (n, x) -> 2^(n/2)*(-1/x^2)^(-n/2)*KummerU(-n/2, 1/2, -1/(2*x^2)):
seq(print(seq(coeff(simplify(p(n, x)), x, k), k = 0..n)), n = 0 ..9);
# Using the exponential generating function:
egf := exp(x + (t*x)^2 / 2): ser := series(egf, x, 12):
seq(print(seq(coeff(n! * coeff(ser, x, n), t, k), k = 0..n)), n = 0..9);

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)];
Table[K[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 25 2023 *)

from functools import cache
@cache
def K(n: int, k: int) -> int:
    if k %  2: return 0
    if n <  3: return 1
    if n == k: return K(n - 1, n - 2)
    return (K(n - 1, k) * n) // (n - k)
for n in range(10): print([K(n, k) for k in range(n + 1)])

Let p(n, x) = 2^(n/2)*(-1/x^2)^(-n/2)*KummerU(-n/2, 1/2, -1/(2*x^2)).
p(n, 1) = A000085(n); p(n, sqrt(2)) = A047974(n); p(n, 2) = A115329(n);
p(2, n) = A002522(n) (n >= 1); p(3, n) = A056107(n) (n >= 1);
p(n, n) = A359739(n) (n >= 1); 2^n*p(n, 1/2) = A005425(n).
K(n, k) = [x^k] p(n, x).
K(n, k) = [t^k] (n! * [x^n] exp(x + (t*x)^2 / 2)).
K(n, n) = A123023(n).
K(n, n-1) = A123023(n + 1).
K(2*n, 2*n) = A001147(n).
K(4*n, 2*n) = A359761, the central terms without zeros.
K(2*n+2, 2*n) = A001879.
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)).
Sum_{k=0..n} Sum_{j=0..k} K(n, j) = A000085(n + 1).
For a recursion see the Python program.

A344911 Concatenated Bessel-scaled Pascal triangles. Irregular triangle read by rows, T(n,k) with n >= 0 and 0 <= k <= (2n(n + 4) - 1 + (-1)^n)/8.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 3, 3, 1, 4, 6, 4, 1, 6, 12, 6, 3, 1, 5, 10, 10, 5, 1, 10, 30, 30, 10, 15, 15, 1, 6, 15, 20, 15, 6, 1, 15, 60, 90, 60, 15, 45, 90, 45, 15, 1, 7, 21, 35, 35, 21, 7, 1, 21, 105, 210, 210, 105, 21, 105, 315, 315, 105, 105, 105

Offset: 0

Peter Luschny, Jun 03 2021

Let p(n) = Sum_{k=0..n/2} Sum_{j=0..n-2*k} (n!/(2^k*k!*j!*(n-2*k-j)!))*x^j*y^(n-2*k-j). Row n of the triangle is defined as the coefficient list of the polynomials p(n), where the monomials are in degree-lexicographic order.
One observes: The triangle of the coefficients appears as a series of concatenated subtriangles. The first one is Pascal's triangle A007318. Appending the rows of triangle A094305 on the right side starts in row 2. In row 4, the next triangle is appended, which is A344565. This scheme goes on indefinitely.
This can be formalized as follows: Let C(n) denote row n of the binomial triangle, set c.C(n) = Seq_{j=0..n} c*binomial(n, j), and let B(n, k) denote the Bessel numbers A100861(n, k). Then T(n) = Seq_{k=0..n/2} B(n, k).C(n-2*k). Since B(n, k) = binomial(n, 2*k)*(2*k - 1)!! it follows that: T(n) = Seq_{k=0..n/2} Seq_{j=0..n-2*k} binomial(n, 2*k)*binomial(n-2*k, j)*(2*k-1)!!. This expression equals the coefficient list of p(n) since the monomials are in degree-lexicographic order.
The polynomials are also the unsigned, probabilist's Hermite polynomials H_n(x+y)
which are discussed in A344678. The coefficients are listed there in a different order which do not reveal the structure described above.

			The triangle begins:
[0] [ 1 ]
[1] [ 1, 1 ]
[2] [ 1, 2,  1 ][ 1 ]
[3] [ 1, 3,  3,   1 ][ 3,   3 ]
[4] [ 1, 4,  6,   4,   1 ][ 6,   12,    6 ][ 3 ]
[5] [ 1, 5, 10,  10,   5,   1 ][ 10,   30,  30, 10 ][ 15, 15 ]
[6] [ 1, 6, 15,  20,  15,   6,    1 ][ 15,  60, 90,   60, 15 ][ 45, 90, 45][ 15 ]
.
With the notations in the comment row 7 concatenates:
B(7, 0).C(7) =   1.[1, 7, 21, 35, 35, 21, 7, 1] = [1, 7, 21, 35, 35, 21, 7, 1],
B(7, 1).C(5) =  21.[1, 5, 10, 10, 5, 1]         = [21, 105, 210, 210, 105, 21],
B(7, 2).C(3) = 105.[1, 3, 3, 1]                 = [105, 315, 315, 105],
B(7, 3).C(1) = 105.[1, 1]                       = [105, 105].
.
p_6(x,y) = x^6 + 6*x^5*y + 15*x^4*y^2 + 20*x^3*y^3 + 15*x^2*y^4 + 6*x*y^5 + y^6 +
15*x^4 + 60*x^3*y + 90*x^2*y^2 + 60*x*y^3 + 15*y^4 + 45*x^2 + 90*x*y + 45*y^2 + 15.

Cf. A005425 (row sums), A100861 (scaling factors).

Cf. A007318, A094305, A099174, A344565, A344678.

P := n -> add(add(n!/(2^k*k!*j!*(n-2*k-j)!)*y^(n-2*k-j)*x^j, j=0..n-2*k), k=0..n/2):
seq(seq(subs(x = 1, y = 1, m), m = [op(P(n))]), n = 0..7);
# Alternatively, without polynomials:
B := (n, k) -> binomial(n, 2*k)*doublefactorial(2*k-1):
C := n -> seq(binomial(n, j), j=0..n):
seq(seq(B(n, k)*C(n-2*k), k = 0..n/2), n = 0..7);
# Based on the e.g.f. of the polynomials:
T := proc(numofrows) local gf, ser, n, m;
gf := exp(t^2/2)*exp(t*(x + y)); ser := series(gf, t, numofrows+1);
for n from 0 to numofrows do [op(sort(n!*expand(coeff(ser, t, n))))];
print(seq(subs(x=1, y=1, m), m = %)) od end: T(7);

P[n_] := Sum[ Sum[n! / (2^k k! j! (n - 2k - j)!) y^(n - 2k - j) x^j, {j, 0, n-2k}], {k, 0, n/2}];
DegLexList[p_] := MonomialList[p, {x, y}, "DegreeLexicographic"] /. x->1 /. y->1;
Table[DegLexList[P[n]], {n, 0, 7}] // Flatten

The bivariate e.g.f. exp(t^2/2)*exp(t*(x + y)) = Sum_{n>=0} H_n(x + y)*t^n/n!, where H_n(x) are the unsigned, modified Hermite polynomials A099174, is given by Tom Copeland in A344678.

A244492 Triangle read by rows: T(n,k) (n>=2, 0 <= k <= n-2) = n!/(2^ii!k!), where k=n-2i (or 0 for entries with wrong parity).

Original entry on oeis.org

1, 0, 3, 3, 0, 6, 0, 15, 0, 10, 15, 0, 45, 0, 15, 0, 105, 0, 105, 0, 21, 105, 0, 420, 0, 210, 0, 28, 0, 945, 0, 1260, 0, 378, 0, 36, 945, 0, 4725, 0, 3150, 0, 630, 0, 45

Offset: 0

N. J. A. Sloane, Jul 05 2014

			Triangle begins:
    1;
    0,   3;
    3,   0,    6;
    0,  15,    0,   10;
   15,   0,   45,    0,   15;
    0, 105,    0,  105,    0,  21;
  105,   0,  420,    0,  210,   0,  28;
    0, 945,    0, 1260,    0, 378,   0, 36;
  945,   0, 4725,    0, 3150,   0, 630,  0, 45;
  ...

J. East and R. D. Gray, Idempotent generators in finite partition monoids and related semigroups, arXiv preprint arXiv:1404.2359 [math.GR], 2014-2016.

This is A099174 without the two rightmost diagonals.

T[n_, k_] := With[{i = (n-k)/2}, If[EvenQ[n-k], n!/(2^i i! k!), 0]];
Table[T[n, k], {n, 2, 10}, {k, 0, n-2}] // Flatten (* Jean-François Alcover, Nov 25 2018 *)

A378100 Number of involutions in the symmetric group S_n with at least one fixed point.

Original entry on oeis.org

0, 1, 1, 4, 7, 26, 61, 232, 659, 2620, 8551, 35696, 129757, 568504, 2255345, 10349536, 44179711, 211799312, 962854399, 4809701440, 23103935021, 119952692896, 605135328337, 3257843882624, 17175956434375, 95680443760576, 525079354619951, 3020676745975552

Offset: 0

Maniru Ibrahim, Nov 16 2024

In other words, a(n) is the number of involutions in S_n that are not derangements.

			a(4) = 7: (1,2)(3)(4), (1,3)(2)(4), (1,4)(2)(3), (1)(2,3)(4), (1)(2,4)(3), (1)(2)(3,4), (1)(2)(3)(4).

Cf. A000085 (involutions), A000166 (derangements), A002467 (permutations with a fixed point), A099174, A123023 (involutions that are derangements).

a := proc(n)
    local k, total, deranged;
    total := add(factorial(n)/(factorial(n-2*k)*2^k*factorial(k)), k=0..floor(n/2));
    if mod(n, 2) = 0 then
        deranged := factorial(n)/(2^(n/2)*factorial(n/2));
    else
        deranged := 0;
    end if;
    return total - deranged;
end proc:
seq(a(n), n=1..20);
# second Maple program:
a:= proc(n) option remember; `if`(n<4, [0, 1$2, 4][n+1],
      a(n-1)+(2*n-3)*a(n-2)-(n-2)*(a(n-3)+(n-3)*a(n-4)))
    end:
seq(a(n), n=0..27);  # Alois P. Heinz, Nov 24 2024

a[n_] := Module[{total, deranged},
  total = Sum[n! / ((n - 2 k)! * 2^k * k!), {k, 0, Floor[n/2]}];
  deranged = If[EvenQ[n], n! / (2^(n/2) * (n/2)!), 0];
  total - deranged
];
Table[a[n], {n, 1, 20}]

my(x='x+O('x^30)); Vec(serlaplace(exp(x+x^2/2)-exp(x^2/2))) \\ Joerg Arndt, Nov 27 2024

from math import factorial
def a(n):
    total = sum(factorial(n) // (factorial(n - 2 * k) * 2**k * factorial(k))
                for k in range(n // 2 + 1))
    deranged = factorial(n) // (2**(n // 2) * factorial(n // 2)) if n % 2 == 0 else 0
    return total - deranged
print([a(n) for n in range(1, 21)])

a(n) = Sum_{k=0..floor(n/2)} n! / ((n-2k)! * 2^k * k!) - (n! / (2^(n/2) * (n/2)!) * (1 - (n mod 2))).
a(n) = A000085(n) - A123023(n).
a(n) = A000085(n) for odd n.
From Alois P. Heinz, Nov 24 2024: (Start)
E.g.f.: exp(x*(2+x)/2)-exp(x^2/2).
a(n) = Sum_{k=1..n} A099174(n,k). (End)

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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!).

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A344911 Concatenated Bessel-scaled Pascal triangles. Irregular triangle read by rows, T(n,k) with n >= 0 and 0 <= k <= (2*n*(n + 4) - 1 + (-1)^n)/8.

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A244492 Triangle read by rows: T(n,k) (n>=2, 0 <= k <= n-2) = n!/(2^i*i!*k!), where k=n-2i (or 0 for entries with wrong parity).

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A378100 Number of involutions in the symmetric group S_n with at least one fixed point.

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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) := (2z)!/(2^zz!).

A344911 Concatenated Bessel-scaled Pascal triangles. Irregular triangle read by rows, T(n,k) with n >= 0 and 0 <= k <= (2n(n + 4) - 1 + (-1)^n)/8.

A244492 Triangle read by rows: T(n,k) (n>=2, 0 <= k <= n-2) = n!/(2^ii!k!), where k=n-2i (or 0 for entries with wrong parity).