A001464 -id:A001464 - 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.

A051684 Auxiliary sequence for calculation of number of even permutations of degree n and order exactly 2.

Original entry on oeis.org

0, -1, -3, -3, 5, 15, -21, -133, 27, 1215, 935, -12441, -23673, 138047, 469455, -1601265, -9112561, 18108927, 182135007, -161934625, -3804634785, -404007681, 83297957567

Offset: 1

Vladeta Jovovic

sign

V. Jovovic, Some combinatorial characteristics of symmetric and alternating groups (in Russian), Belgrade, 1980, unpublished.

Cf. A001189, A051685.

a(n) = c(n, 2), where c(n, d)=Sum_{k=1..n} (-1)^(k+1)*(n-1)!/(n-k)! *Sum_{l:lcm{k, l}=d} c(n-k, l), c(0, 1)=1.
a(n)=2*A048099(n)-A001189(n)=A048099(n)-A001465(n) a(n)=(-1)^n*A001464(n)-1 a(n)=a(n-1)-(n-1)*(a(n-2)+1) E.g.f.: -e^x+e^(x-(1/2)*x^2) - Matthew J. White (mattjameswhite(AT)hotmail.com), Mar 02 2006
a(n) = Sum((-1)^j*n!/(2^j*j!*(n-2*j)!),j=1..floor(n/2)). - Vladeta Jovovic, Mar 06 2006

A104548 Triangle read by rows giving coefficients of Bessel polynomial p_n(x).

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 1, 3, 3, 0, 1, 6, 15, 15, 0, 1, 10, 45, 105, 105, 0, 1, 15, 105, 420, 945, 945, 0, 1, 21, 210, 1260, 4725, 10395, 10395, 0, 1, 28, 378, 3150, 17325, 62370, 135135, 135135, 0, 1, 36, 630, 6930, 51975, 270270, 945945, 2027025, 2027025, 0

Offset: 0

Eric W. Weisstein, Mar 14 2005

			Bessel polynomials begin with:
      x;
      x +     x^2;
    3*x +   3*x^2 +    x^3;
   15*x +  15*x^2 +  6*x^3 +    x^4;
  105*x + 105*x^2 + 45*x^3 + 10*x^4 + x^5;
  ...
Triangle of coefficients begins as:
  0;
  1,  0;
  1,  1    0;
  1,  3,   3     0;
  1,  6,  15,   15      0;
  1, 10,  45,  105,   105      0;
  1, 15, 105,  420,   945,   945       0;
  1, 21, 210, 1260,  4725, 10395,  10395       0;
  1, 28, 378, 3150, 17325, 62370, 135135, 135135    0;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Eric Weisstein's World of Mathematics, Bessel Polynomial

Essentially the same as A001498 (the main entry).

Cf. A000085, A000806, A001464, A001515.

A104548:= func< n,k | k eq n select 0 else Binomial(n-1,k)*Factorial(n+k-1)/(2^k*Factorial(n-1)) >;
[A104548(n,k): k in [0..n], n in [0..13]]; // G. C. Greubel, Jan 02 2023

T[n_, k_]:= If[k==n, 0, Binomial[n-1,k]*(n+k-1)!/(2^k*(n-1)!)];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 02 2023 *)

def A104548(n,k): return 0 if (k==n) else binomial(n-1,k)*factorial(n+k-1)/(2^k*factorial(n-1))
flatten([[A104548(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jan 02 2023

From G. C. Greubel, Jan 02 2023: (Start)
T(n, k) = binomial(n-1,k)*(n+k-1)!/(2^k*(n-1)!), with T(n, n) = 0.
Sum_{k=0..n} T(n, k) = A001515(n-1).
Sum_{k=0..n} (-1)^k*T(n, k) = A000806(n-1).
Sum_{k=0..floor(n/2)} T(n-k, k) = A000085(n-1).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A001464(n-1). (End)

T(0, 0) = 0 prepended by G. C. Greubel, Jan 02 2023

A328141 a(n) = a(n-1) - (n-2)*a(n-2), with a(0)=1, a(1)=2.

Original entry on oeis.org

1, 2, 2, 0, -4, -4, 12, 32, -40, -264, 56, 2432, 1872, -24880, -47344, 276096, 938912, -3202528, -18225120, 36217856, 364270016, -323869248, -7609269568, -808015360, 166595915136, 185180268416, -3813121694848, -8442628405248, 90698535660800, 318649502602496, -2220909495899904

Offset: 0

G. C. Greubel, Oct 04 2019

sign

Former title and formula of A122033, but not the data.

G. C. Greubel, Table of n, a(n) for n = 0..800

Cf. A001464, A060821, A121966, A122033.

a:=[1,2];; for n in [3..35] do a[n]:=a[n-1]-(n-3)*a[n-2]; od; a;

I:=[1,2]; [n le 2 select I[n] else Self(n-1) - (n-3)*Self(n-2): n in [1..35]];

a:= proc (n) option remember;
if n < 2 then n+1
else a(n-1) - (n-2)*a(n-2)
fi;
end proc; seq(a(n), n = 0..35);

a[n_]:= a[n]= If[n<2, n+1, a[n-1]-(n-2)*a[n-2]]; Table[a[n], {n,0,35}]

my(m=35, v=concat([1,2], vector(m-2))); for(n=3, m, v[n] = v[n-1] - (n-3)*v[n-2] ); v

def a(n):
    if n<2: return n+1
    else: return a(n-1) - (n-2)*a(n-2)
[a(n) for n in (0..35)]

a(n) = a(n-1) - (n-2)*a(n-2), with a(0)=1, a(1)=2.
E.g.f.: 1 + sqrt(2*e*Pi)*( erf(1/sqrt(2)) + erf((x-1)/sqrt(2)) ), where erf(x) is the error function.
a(n) = 2*(-1)^(n-1)*A001464(n-1).
a(n) = 2*(1/sqrt(2))^(n-1) * Hermite(n-1, 1/sqrt(2)), n > 0.

A369755 Expansion of e.g.f. exp( (1 - (1+x)^4)/4 ).

Original entry on oeis.org

1, -1, -2, 2, 28, 44, -464, -3088, 1408, 135872, 726976, -2959936, -67261952, -293413888, 3054389248, 52458520064, 178569842176, -3909868400128, -60465254054912, -149165881689088, 6569005278939136, 98054837101881344, 158559568611401728, -14356527387138039808

Offset: 0

Seiichi Manyama, Jan 31 2024

sign

Eric Weisstein's World of Mathematics, Bell Polynomial.

Cf. A001464, A252284, A369756.

Cf. A049426.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((1-(1+x)^4)/4)))

a(0) = 1; a(n) = -(n-1)! * Sum_{k=1..min(4,n)} binomial(3,k-1) * a(n-k)/(n-k)!.
a(n) = Sum_{k=0..n} 4^k * Stirling1(n,k) * Bell_k(-1/4), where Bell_n(x) is n-th Bell polynomial.
D-finite with recurrence a(n) +a(n-1) +3*(n-1)*a(n-2) +3*(n-1)*(n-2)*a(n-3) +(n-1)*(n-2)*(n-3)*a(n-4)=0. - R. J. Mathar, Feb 02 2024

A369756 Expansion of e.g.f. exp( (1 - (1+x)^5)/5 ).

Original entry on oeis.org

1, -1, -3, -1, 49, 255, -275, -13105, -83775, 170495, 8290045, 69257055, -111005135, -9684015745, -109196883795, -31470300625, 17728458119425, 276531029694975, 904537471692925, -44728487203650625, -1000823562359108175, -7110596979389965825

Offset: 0

Seiichi Manyama, Jan 31 2024

sign

Eric Weisstein's World of Mathematics, Bell Polynomial.

Cf. A001464, A252284, A369755.

Cf. A049427, A369753.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((1-(1+x)^5)/5)))

a(0) = 1; a(n) = -(n-1)! * Sum_{k=1..min(5,n)} binomial(4,k-1) * a(n-k)/(n-k)!.

a(n) = Sum_{k=0..n} 5^k * Stirling1(n,k) * Bell_k(-1/5), where Bell_n(x) is n-th Bell polynomial.

A014775 Expansion of exp ( - x - (1/2)x^2 - (1/6)x^3).

Original entry on oeis.org

1, -1, 0, 1, 2, -6, -14, 20, 204, 28, -2584, -6876, 33760, 219296, -121848, -6020456, -15177904, 126126960, 950679424, -898745392, -38731873824, -123922308896, 1126028191520, 10547325457536, -5093629711808

Offset: 0

N. J. A. Sloane

sign

Cf. A001464, A001680.

With[{nmax = 30}, CoefficientList[Series[Exp[-x - x^2/2 - x^3/6], {x, 0, nmax}], x] * Range[0, nmax]!] (* Vaclav Kotesovec, Feb 09 2020 *)

A085386 E.g.f. cosh(x+x^2/2).

Original entry on oeis.org

1, 0, 1, 3, 4, 10, 46, 126, 316, 1296, 5356, 17380, 63856, 296088, 1264264, 4940040, 22302736, 110455936, 507711376, 2313783216, 11798364736, 61878663840, 309240315616, 1587272962528, 8792390355904, 48793502304000

Offset: 0

Paul Barry, Jun 27 2003

a(n) is the number of involutions with an even number of cycles. - Geoffrey Critzer, Mar 17 2013

Cf. A085387.

nn=25;Range[0,nn]!CoefficientList[Series[Cosh[x+x^2/2],{x,0,nn}],x]  (* Geoffrey Critzer, Mar 17 2013 *)

a(n)=(A000085(n)+A001464(n))/2

A266503 Number of symmetric difference-closed 4-sets consisting of sets consisting of an even number of pairwise disjoint 2-subsets of {1,2,...,n}.

Original entry on oeis.org

0, 0, 0, 0, 0, 15, 105, 735, 4095, 26775, 162855, 1105335, 7187895, 51126075, 356831475, 2676468795, 19890931515, 156769986555, 1232704469115, 10178240218875, 84190426730235, 725667326178795

Offset: 1

John M. Campbell, Jan 24 2016

A set of this form forms a group (isomorphic to the Klein four-group) under the symmetric difference operation. Such sets may be regarded in a natural way as Klein four-subgroups of the alternating group A_n.

			For example, there are a(n) = 15 sets of this form in the case whereby n=6:
{{{12}, {34}}, {{34}, {56}}, {{12}, {56}}, {}}
{{{12}, {35}}, {{35}, {46}}, {{12}, {46}}, {}}
{{{12}, {36}}, {{36}, {45}}, {{12}, {45}}, {}}
{{{13}, {24}}, {{24}, {56}}, {{13}, {56}}, {}}
{{{13}, {25}}, {{25}, {46}}, {{13}, {46}}, {}}
{{{13}, {26}}, {{26}, {45}}, {{13}, {45}}, {}}
{{{14}, {23}}, {{23}, {56}}, {{14}, {56}}, {}}
{{{14}, {25}}, {{25}, {36}}, {{14}, {36}}, {}}
{{{14}, {26}}, {{26}, {35}}, {{14}, {35}}, {}}
{{{15}, {23}}, {{23}, {46}}, {{15}, {46}}, {}}
{{{15}, {24}}, {{24}, {36}}, {{15}, {36}}, {}}
{{{15}, {26}}, {{26}, {34}}, {{15}, {34}}, {}}
{{{16}, {23}}, {{23}, {45}}, {{16}, {45}}, {}}
{{{16}, {24}}, {{24}, {35}}, {{16}, {35}}, {}}
{{{16}, {25}}, {{25}, {34}}, {{16}, {34}}, {}}

John Campbell, A class of symmetric difference-closed sets related to commuting involutions, Discrete Mathematics & Theoretical Computer Science, Vol 19 no. 1, 2017.

Cf. A267840.

a[n_] := n!*Sum[Sum[Sum[(2^(k-2*i-2*j))/(k!*(2*i-k)!*(2*j-k)!*(n-4*i-4*j+2*k)!*(KroneckerDelta[i, j]+KroneckerDelta[i, k]+1)!), {k, Max[i, 2*i+2*j-Floor[n/2]], Min[2*j, Floor[1/4*(4*i+4*j-1)]]}], {j, 1, i}], {i, 1, Floor[n/2]}] ; Print[Table[a[n], {n, 1, 22}]] ;
Rest[CoefficientList[Series[E^x/3 - E^(-x*(x-2)/2)/8 - E^(x*(x+2)/2)/4 + E^(x*(3*x+2)/2)/24, {x, 0, 30}], x] * Range[0, 30]!] (* Vaclav Kotesovec, Apr 10 2016 *)

a(n) = n!*sum(sum(sum((2^(k-2*i-2*j))/(k!*(2*i-k)!*(2*j-k)!*(n-4*i-4*j+2*k)!*(delta(i, j)+delta(i, k)+1)!), k=max(i, 2*i+2*j-[n/2])..min(2*j, [1/4*(4*i+4*j-1)])), j=1..i), i=1..[n/2]).
From Vaclav Kotesovec, Apr 10 2016: (Start)
Recurrence: (n-6)*(n-4)*(n-2)*a(n) = (2*n - 7)*(2*n^2 - 14*n + 15)*a(n-1) + 3*(n-7)*(n-1)*(n^2 - 7*n + 11)*a(n-2) - (n-2)*(n-1)*(9*n^2 - 85*n + 189)*a(n-3) + (n-3)*(n-2)*(n-1)*(n^2 - n - 22)*a(n-4) - 2*(n-4)^2*(n-3)*(n-2)*(n-1)*a(n-5) - (n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(3*n - 19)*a(n-6) + 3*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-7).
E.g.f.: exp(x)/3 - exp(-x*(x-2)/2)/8 - exp(x*(x+2)/2)/4 + exp(x*(3*x+2)/2)/24.
a(n) ~ 2^(-7/2) * 3^(n/2 - 1) * exp(sqrt(n/3) - n/2 - 1/12) * n^(n/2).
(End)
a(n) = 1/3 + (-1)^n*A001464(n)/8 - A000085(n)/4 + A115327(n)/24. - Vaclav Kotesovec, May 28 2016

A295289 Expansion of 1/(1 + x + x^2/(1 + 2x + x^2/(1 + 3x + x^2/(1 + 4*x + x^2/(1 + ...))))), a continued fraction.

Original entry on oeis.org

1, -1, 0, 3, -8, 9, 19, -141, 448, -759, -951, 14355, -71810, 238617, -421622, -1283181, 17699702, -110524503, 494858579, -1480719213, 12947578, 41684591673, -396443074054, 2562153015315, -12548536326806, 39305562343881, 52542301031459, -2158744047173613

Offset: 0

Ilya Gutkovskiy, Nov 19 2017

sign

Cf. A000110, A001464, A006789.

nmax = 27; CoefficientList[Series[1/(1 + x + ContinuedFractionK[x^2, 1 + (k + 1) x, {k, 1, nmax}]), {x, 0, nmax}], x]

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

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A051684 Auxiliary sequence for calculation of number of even permutations of degree n and order exactly 2.

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A104548 Triangle read by rows giving coefficients of Bessel polynomial p_n(x).

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A328141 a(n) = a(n-1) - (n-2)*a(n-2), with a(0)=1, a(1)=2.

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A369755 Expansion of e.g.f. exp( (1 - (1+x)^4)/4 ).

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PARI

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A369756 Expansion of e.g.f. exp( (1 - (1+x)^5)/5 ).

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PARI

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A014775 Expansion of exp ( - x - (1/2)*x^2 - (1/6)*x^3).

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Mathematica

A085386 E.g.f. cosh(x+x^2/2).

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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!).

A014775 Expansion of exp ( - x - (1/2)x^2 - (1/6)x^3).

A295289 Expansion of 1/(1 + x + x^2/(1 + 2x + x^2/(1 + 3x + x^2/(1 + 4*x + x^2/(1 + ...))))), a continued fraction.