A005425 -id:A005425 - cp's OEIS frontend

A094072 Coefficients arising in combinatorial field theory.

1, 6, 50, 615, 10192, 214571, 5544394, 171367020, 6208928376, 259542887975, 12356823485580, 662921411131909, 39714830070598204, 2636484537372437498, 192653800829700013970, 15405383160836582657251

Offset: 0

N. J. A. Sloane, May 01 2004

nonn

P. Blasiak, K. A. Penson, A. I. Solomon, A. Horzela and G. E. H. Duchamp, Some useful combinatorial formulas for bosonic operators, J. Math. Phys. 46, 052110 (2005) (6 pages).

P. Blasiak, K. A. Penson, A. I. Solomon, A. Horzela and G. E. H. Duchamp, Combinatorial field theories via boson normal ordering, arXiv:quant-ph/0405103, 2004-2006. The title of this paper in the arXiv was later changed to "Some useful combinatorial formulas for bosonic operators"

Cf. A000085, A005425, A094070, A094071, A094073, A094074.

Cf. A000110.

with(combinat): seq(bell(n+1)*sum(k^(n+1-k)*binomial(n+1,k),k=1..n+1),n=0..18);

Table[BellB[n+1]Sum[Binomial[n+1,k]k^(n+1-k),{k,n+1}],{n,0,20}] (* Harvey P. Dale, Feb 05 2015 *)

a(n) = B(n+1)*Sum_{k=1..n+1} binomial(n+1, k)*k^(n+1-k), where B(n) are the Bell numbers (A000110). - Emeric Deutsch, Nov 23 2004

E.g.f.: exp(-1)*Sum_{k>=0} exp(k*x*exp(k*x))/k!. - Vladeta Jovovic, Sep 26 2006

More terms from Emeric Deutsch, Nov 23 2004

A246072 Number A(n,k) of permutations p on [2n] satisfying p^k(i) = i for all i in [n]; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 2, 1, 1, 24, 1, 2, 2, 720, 1, 1, 10, 6, 40320, 1, 2, 10, 84, 24, 3628800, 1, 1, 16, 108, 1032, 120, 479001600, 1, 2, 2, 264, 1800, 17040, 720, 87178291200, 1, 1, 18, 150, 6672, 47520, 359280, 5040, 20922789888000

Offset: 0

Alois P. Heinz, Aug 12 2014

Column k=2 is n! * A005425(n), column k=3 is n! * A242054(n). - Vaclav Kotesovec, Aug 13 2014

			A(2,3) = 10: (1,2,3,4), (1,2,4,3), (1,3,4,2), (1,4,2,3), (2,3,1,4), (2,4,3,1), (3,1,2,4), (3,2,4,1), (4,1,3,2), (4,2,1,3).
a(2,4) = 16: (1,2,3,4), (1,2,4,3), (1,3,2,4), (1,4,3,2), (2,1,3,4), (2,1,4,3), (2,3,4,1), (2,4,1,3), (3,1,4,2), (3,2,1,4), (3,4,1,2), (3,4,2,1), (4,1,2,3), (4,2,3,1), (4,3,1,2), (4,3,2,1).
A(2,5) = 2: (1,2,3,4), (1,2,4,3).
A(3,1) = 6: (1,2,3,4,5,6), (1,2,3,4,6,5), (1,2,3,5,4,6), (1,2,3,5,6,4), (1,2,3,6,4,5), (1,2,3,6,5,4).
Square array A(n,k) begins:
0 :        1,   1,     1,     1,      1,      1, ...
1 :        2,   1,     2,     1,      2,      1, ...
2 :       24,   2,    10,    10,     16,      2, ...
3 :      720,   6,    84,   108,    264,    150, ...
4 :    40320,  24,  1032,  1800,   6672,   2424, ...
5 :  3628800, 120, 17040, 47520, 241440, 109200, ...

Alois P. Heinz, Antidiagonals n = 0..90, flattened

Columns k=0-1 give: A010050, A000142. Main diagonal gives A246073.

Cf. A005425, A242054, A246070 (the same for endofunctions).

with(numtheory): with(combinat): M:=multinomial:
b:= proc(n, k, p) local l, g; l, g:= sort([divisors(p)[]]),
      proc(k, m, i, t) option remember; local d, j; d:= l[i];
        `if`(i=1, m!, add(M(k, k-(d-t)*j, (d-t)$j)/j!*
         (d-1)!^j *M(m, m-t*j, t$j) *g(k-(d-t)*j, m-t*j,
        `if`(d-t=1, [i-1, 0], [i, t+1])[]), j=0..min(k/(d-t),
        `if`(t=0, [][], m/t))))
      end; g(k, n-k, nops(l), 0)
    end:
A:= (n, k)-> `if`(k=0, (2*n)!, b(2*n, n, k)):
seq(seq(A(n, d-n), n=0..d), d=0..12);

multinomial[n_, k_List] := n!/Times @@ (k!); M = multinomial; b[n_, k_, p_] := b[n, k, p] = Module[{l, g}, l = Sort[Divisors[p]]; g[k0_, m_, i_, t_] := g[k0, m, i, t] = Module[{d}, d = l[[i]]; If[i == 1, m!, Sum[ M[k0, Join[{k0-(d-t)*j}, Table[d-t, {j}]]]/j!*(d-1)!^j*M[m, Join[{m-t*j}, Table[t, {j}]]]*If[d-t == 1, g[k0-(d-t)*j, m-t*j, i-1, 0], g[k0-(d-t)*j, m-t*j, i, t+1]], {j, 0, Min[k0/(d-t), If[t == 0, Infinity, m/t]]}]]]; g[k, n-k, Length[l], 0]]; A[n_, k_] := If[k == 0, (2*n)!, b[2*n, n, k]]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Jan 06 2015, after Alois P. Heinz *)

A094070 a(n) = A000085(n) * A000110(n).

Original entry on oeis.org

1, 4, 20, 150, 1352, 15428, 203464, 3162960, 55405140, 1101298600, 24222234720, 590544046744, 15715973012248, 456341011254560, 14312979247985120, 484253161428902192, 17550722413456774848, 680244627812139042016, 28053748582811428182080, 1228896901162555453603712

Offset: 0

N. J. A. Sloane, May 01 2004

nonn

Coefficients arising in combinatorial field theory.

P. Blasiak, K. A. Penson, A. I. Solomon, A. Horzela and G. E. H. Duchamp, Some useful combinatorial formulas for bosonic operators, J. Math. Phys. 46, 052110 (2005) (6 pages).
P. Blasiak, K. A. Penson, A. I. Solomon, A. Horzela and G E. H. Duchamp, Combinatorial field theories via boson normal ordering, preprint, Apr 27 2004.

P. Blasiak, K. A. Penson, A. I. Solomon, A. Horzela and G. E. H. Duchamp, Combinatorial field theories via boson normal ordering, arXiv:quant-ph/0405103, 2004-2006. The title of this paper in the arXiv was later changed to "Some useful combinatorial formulas for bosonic operators"
A. Horzela, P. Blasiak, G. E. H. Duchamp, K. A. Penson and A. I. Solomon, A product formula and combinatorial field theory

Cf. A000085, A005425, A094071, A094072, A094073, A094074.

with(combinat): with(orthopoly): seq((I/sqrt(2))^(n+1)*H(n+1,-I/sqrt(2))*bell(n+1),n=0..17); # Emeric Deutsch, Nov 22 2004

a[n_] := Sum[StirlingS1[n+1, k] 2^k BellB[k, 1/2], {k, 0, n+1}] BellB[n+1];
Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Aug 07 2018 *)

a(n) = (i/sqrt(2))^(n+1)*H(n+1, -i/sqrt(2))*Bell(n+1), where i=sqrt(-1), H(n, x) are the Hermite polynomials and Bell(n) are the Bell numbers. - Emeric Deutsch, Nov 22 2004

More terms from Ralf Stephan, Oct 14 2004

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!).

Original entry on oeis.org

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.

A027412 a(n) = 2a(n-1) + (n-2)a(n-2) with a(0) = 1, a(1) = 2.

Original entry on oeis.org

1, 2, 4, 10, 28, 86, 284, 998, 3700, 14386, 58372, 246218, 1076156, 4860710, 22635292, 108459814, 533813716, 2694524642, 13930068740, 73667056394, 398075350108, 2195824771702, 12353156545564, 70818633296870

Offset: 0

Olivier Gérard

Vincenzo Librandi, Table of n, a(n) for n = 0..200
R. Donaghey, Binomial self-inverse sequences and tangent coefficients, J. Combin. Theory, Series A, 21 (1976), 155-163.

Cf. A005425.

[n le 2 select n else 2*Self(n-1) + (n-3)*Self(n-2): n in [1..40]]; // G. C. Greubel, Apr 01 2023

RecurrenceTable[{a[0]==1, a[1]==2, a[n]==2 a[n-1] + (n-2) a[n-2]}, a,{ n, 40}] (* Vincenzo Librandi, Dec 29 2012 *)
Table[If[n==0, 1, 2*(-I/Sqrt[2])^(n-1)*HermiteH[n-1, I*Sqrt[2]]], {n, 0, 40}] (* G. C. Greubel, Apr 01 2023 *)

[1]+[2*(-i/sqrt(2))^(n-1)*hermite(n-1, i*sqrt(2)) for n in range(1,41)] # G. C. Greubel, Apr 01 2023

a(n) = 2 * A005425(n-1), n > 0.

a(n) ~ sqrt(2)*exp(2*sqrt(n)-n/2-1)*n^(n/2-1/2). - Vaclav Kotesovec, Oct 08 2012

A085387 E.g.f exp(x)*cosh(x+x^2/2).

Original entry on oeis.org

1, 1, 2, 7, 23, 71, 242, 925, 3649, 14593, 61082, 269039, 1220375, 5658823, 27047386, 133453429, 674644673, 3482517185, 18399534386, 99518837527, 549283557463, 3088289136391, 17697783668930, 103351677649037, 614068613826689

Offset: 0

Paul Barry, Jun 27 2003

Binomial transform of A085386. A085387 is the mean of A005425 and the sequence with e.g.f. exp(-x^2/2).

Cf. A001147.

CoefficientList[Series[E^x*Cosh[x+x^2/2], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 19 2013 *)

E.g.f.: exp(x)*cosh(x+x^2/2).
Conjecture: a(n) -3*a(n-1) +2*a(n-2) +(-2*n+3)*a(n-3) +-(n-3)^2*a(n-4) +(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Nov 24 2012
a(n) ~ 2^(-3/2) * exp(2*sqrt(n) - n/2 - 1) * n^(n/2) * (1+5/(6*sqrt(n))). - Vaclav Kotesovec, Oct 19 2013

A238803 Number of ballot sequences of length 2n with exactly n fixed points.

Original entry on oeis.org

1, 1, 3, 9, 29, 99, 357, 1351, 5343, 21993, 93923, 414969, 1892277, 8887291, 42912261, 212676951, 1080355463, 5617772049, 29868493827, 162204146857, 898874710797, 5078665886931, 29232738375653, 171294038649639, 1021117638212079, 6188701520663929

Offset: 0

Joerg Arndt and Alois P. Heinz, Mar 05 2014

The fixed points are in the positions 1,2,...,n.
Also the number of standard Young tableaux with 2n cells where n is the length of the maximal consecutive sequence 1,2,...,k in the first column.  An alternate definition uses the first row.
All terms are odd because the counted structures come in pairs with exactly one exception.
Except for a(0), first differences of A005425. - Ivan N. Ianakiev, Sep 01 2019

			For n=3 we have the following a(3) = 9 ballot sequences: [1,2,3,1,1,1], [1,2,3,1,2,3], [1,2,3,1,1,2], [1,2,3,1,2,1], [1,2,3,1,4,1], [1,2,3,1,4,2], [1,2,3,1,1,4], [1,2,3,1,2,4], [1,2,3,1,4,5].
Their corresponding tableaux are:
: 1456  14 : 145  146 : 146  14 : 145  14 : 14 :
: 2     25 : 26   25  : 2    26 : 2    25 : 2  :
: 3     36 : 3    3   : 3    3  : 3    3  : 3  :
:          :          : 5    5  : 6    6  : 5  :
:          :          :         :         : 6  :

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 0..750
Wikipedia, Young tableau

Cf. A000085, A005425, A238802.

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

RecurrenceTable[{a[0]==a[1]==1,a[n]==((2n-1)a[n-1]+n(n-2)a[n-2])/(n-1)},a,{n,30}] (* Harvey P. Dale, Jun 25 2014 *)

a(n) = ((2*n-1)*a(n-1)+n*(n-2)*a(n-2))/(n-1) for n>1, a(0) = a(1) = 1.
a(n) = A238802(2*n,n).
a(n) = Sum_{k=0..n} C(n-1,k) * A000085(n-k).
a(n) ~ exp(2*sqrt(n)-n/2-1) * n^(n/2) / sqrt(2) * (1 - 1/(6*sqrt(n))). - Vaclav Kotesovec, Mar 07 2014

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.

A344912 Irregular triangle read by rows, Trow(n) = Seq_{k=0..n/3} Seq_{j=0..n-3k} (n! binomial(n - 3k, j)) / (k!(n - 3k)!3^k).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 2, 1, 4, 6, 4, 1, 8, 8, 1, 5, 10, 10, 5, 1, 20, 40, 20, 1, 6, 15, 20, 15, 6, 1, 40, 120, 120, 40, 40, 1, 7, 21, 35, 35, 21, 7, 1, 70, 280, 420, 280, 70, 280, 280, 1, 8, 28, 56, 70, 56, 28, 8, 1, 112, 560, 1120, 1120, 560, 112, 1120, 2240, 1120

Offset: 0

Peter Luschny, Jun 04 2021

Consider a sequence of Pascal tetrahedrons (depending on a parameter m >= 1), where the slices of the pyramid are scaled. They are given by the e.g.f.s exp(t^m / m) * exp(t*(x + y)), which provide a sequence of bivariate polynomials in x and y, whose monomials are to be ordered in degree-lexicographic order. For m = 1 one gets A109649 (resp. A046816), for m = 2 one gets A344911 (resp. A344678), and for m = 3 the current triangle. The row sums have an unexpected interpretation in A336614 (see the link).

			Triangle begins:
[0] 1;
[1] 1, 1;
[2] 1, 2,  1;
[3] 1, 3,  3,  1,  2;
[4] 1, 4,  6,  4,  1,  8,  8;
[5] 1, 5, 10, 10,  5,  1, 20, 40,  20;
[6] 1, 6, 15, 20, 15,  6,  1, 40, 120, 120,  40,  40;
[7] 1, 7, 21, 35, 35, 21,  7,  1,  70, 280, 420, 280, 70, 280, 280.
.
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 + 40*x^3 + 120*x^2*y + 120*x*y^2 + 40*y^3 + 40.

mjqxxxx, Proof of conjectured formulas for A336614, Mathematics Stack Exchange.

m=1: A109649, (A046816) [row sums A000244], scaling A007318 [row sums A000079].
m=2: A344911, (A344678) [row sums A005425], scaling A100861 [row sums A000085].
m=3: this triangle      [row sums A336614], scaling A118931 [row sums A001470].

B := (n, k) -> n!/(k!*(n - 3*k)!*(3^k)): C := n -> seq(binomial(n, j), j=0..n):
T := (n, k) -> B(n, k)*C(n - 3*k): seq(seq(T(n, k), k = 0..n/3), n = 0..8);

gf := Exp[t^3 / 3] Exp[t (x + y)]; ser := Series[gf, {t, 0, 9}];
P[n_] := Expand[n! Coefficient[ser, t, n]];
DegLexList[p_] := MonomialList[p, {x, y}, "DegreeLexicographic"] /. x->1 /. y->1;
Table[DegLexList[P[n]], {n, 0, 7}] // Flatten

A188144 Binomial transform A140456(n+1) (indecomposable involutions).

Original entry on oeis.org

1, 2, 6, 20, 74, 292, 1218, 5308, 24034, 112484, 542346, 2686268, 13639226, 70863652, 376208706, 2038335580, 11259474754, 63353211332, 362819139978, 2113410084188, 12513610048154, 75274067489284, 459782361574146, 2850369932150908, 17926893505949986, 114337654086861092

Offset: 0

Groux Roland, Mar 22 2011

nonn

a(n) is also the INVERTi transform of A005425(n+1) (self-inverse partial permutations) starting at n=2.

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A005425, A140456.

b:= proc(n) b(n):= `if`(n<2, n+1,  2*b(n-1) + (n-1)*b(n-2)) end:
g:= proc(n) g(n):= `if`(n<1, -1, -add(g(n-i)*b(i), i=1..n)) end:
a:= n-> g(n+2):
seq(a(n), n=0..28);  # Alois P. Heinz, Mar 19 2020

nmax = 18; A140456 = CoefficientList[ Series[1 - 1/Total[ CoefficientList[ Series[Exp[x^2/2 + x], {x, 0, nmax + 2}], x]*Range[0, nmax + 2]!* x^Range[0, nmax + 2]], {x, 0, nmax + 2}], x]; a[n_] := Sum[ Binomial[n, k]*A140456[[k + 3]], {k, 0, n}]; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Jul 03 2013 *)

seq(n)={Vec(1 - 2*x - 1/serlaplace(exp( 2*x + x^2/2 + O(x^3*x^n) )))} \\ Andrew Howroyd, Jan 06 2020

a(n) is the moment of order n for the probability density function: sqrt(2/Pi^3)*exp((x-2)^2/2)/(1+(erfi((x-2)/sqrt(2)))^2) over the interval -infinity..infinity, with erfi the imaginary error function.

G.f.: A(x) = (1 - 2*x - G(0))/x^2; G(k) = 1 - 2*x - x^2*(k+1)/G(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Jan 26 2012

Terms a(19) and beyond from Andrew Howroyd, Jan 06 2020

cp's OEIS Frontend

A094072 Coefficients arising in combinatorial field theory.

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A246072 Number A(n,k) of permutations p on [2n] satisfying p^k(i) = i for all i in [n]; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A094070 a(n) = A000085(n) * A000110(n).

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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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A027412 a(n) = 2*a(n-1) + (n-2)*a(n-2) with a(0) = 1, a(1) = 2.

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A085387 E.g.f exp(x)*cosh(x+x^2/2).

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A238803 Number of ballot sequences of length 2n with exactly n fixed points.

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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!).

A027412 a(n) = 2a(n-1) + (n-2)a(n-2) with a(0) = 1, a(1) = 2.

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.

A344912 Irregular triangle read by rows, Trow(n) = Seq_{k=0..n/3} Seq_{j=0..n-3k} (n! binomial(n - 3k, j)) / (k!(n - 3k)!3^k).