A005425 -id:A005425 - cp's OEIS frontend

A128227 Right border (1,1,1,...) added to A002260.

1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 3, 4, 1, 1, 2, 3, 4, 5, 1, 1, 2, 3, 4, 5, 6, 1, 1, 2, 3, 4, 5, 6, 7, 1, 1, 2, 3, 4, 5, 6, 7, 8, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1

Offset: 0

Gary W. Adamson, Feb 19 2007

Row sums = A000124: (1, 2, 4, 7, 11, 16, ...). n* each term of the triangle gives A128228, having row sums A006000: (1, 4, 12, 28, 55, ...).
Eigensequence of the triangle = A005425: (1, 2, 5, 14, 43, ...). - Gary W. Adamson, Aug 27 2010
From Franck Maminirina Ramaharo, Aug 25 2018: (Start)
T(n,k) is the number of binary words of length n having k letters 1 such that no 1's lie between any pair of 0's.
Let n lines with equations y = (i - 1)*x - (i - 1)^2, i = 1..n, be drawn in the Cartesian plane. For each line, call the half plane containing the point (-1,1) the upper half plane and the other half the lower half-plane. Then T(n,k) is the number of regions that are the intersections of k upper half-planes and n-k lower half-planes. Here, T(0,0) = 1 corresponds to the plane itself. A region obtained from this arrangement of lines can be associated with a length n binary word such that the i-th letter indicates whether the region is located at the i-th upper half-plane (letter 1) or at the lower half-plane (letter 0).
(End)

			First few rows of the triangle are:
1;
1, 1;
1, 2, 1;
1, 2, 3, 1;
1, 2, 3, 4, 1;
1, 2, 3, 4, 5, 1;
1, 2, 3, 4, 5, 6, 1;
1, 2, 3, 4, 5, 6, 7, 1;
1, 2, 3, 4, 5, 6, 7, 8, 1;
...
From _Franck Maminirina Ramaharo_, Aug 25 2018: (Start)
For n = 5, the binary words are
(k = 0) 00000;
(k = 1) 10000, 00001;
(k = 2) 11000, 10001, 00011;
(k = 3) 11100, 11001, 10011, 00111;
(k = 4) 11110, 11101, 11011, 10111, 01111;
(k = 5) 11111.
(End)

A. Bogolmony, Number of Regions N Lines Divide Plane.
Eric Weisstein's World of Mathematics, Plane Division by Lines.
J. E. Wetzel, On the division of the plane by lines, The American Mathematical Monthly Vol. 85 (1978), 647-656.

Cf. A002260, A128228, A000124, A006000, A318274.

Cf. A005425. - Gary W. Adamson, Aug 27 2010

(* first n rows of the triangle *)
a128227[n_] := Table[If[r==q, 1, q], {r, 1, n}, {q, 1, r}]
Flatten[a128227[13]] (* data *)
TableForm[a128227[5]] (* triangle *)
(* Hartmut F. W. Hoft, Jun 10 2017 *)

T(n, k) := if n = k then 1 else k + 1$
for n:0 thru 10 do print(makelist(T(n, k), k, 0, n)); /* Franck Maminirina Ramaharo, Aug 25 2018 */

def T(n, k): return 1 if n==k else k
for n in range(1, 11): print([T(n, k) for k in range(1, n + 1)]) # Indranil Ghosh, Jun 10 2017

from math import comb, isqrt
def A128227(n): return n-comb(r:=(m:=isqrt(k:=n+1<<1))+(k>m*(m+1))+1,2)+(2 if k==m*(m+1) else r) # Chai Wah Wu, Nov 09 2024

"1" added to each row of "start counting again": (1; 1,2; 1,2,3,...) such that a(1) = 1, giving: (1; 1,1; 1,2,1;...).
T(n,k) = k if 1<=kHartmut F. W. Hoft, Jun 10 2017
From Franck Maminirina Ramaharo, Aug 25 2018: (Start)
The n-th row are the coefficients in the expansion of ((x^2 + (n - 2)*x - n)*x^n + 1)/(x - 1)^2.
G.f. for column k: ((k*x + 1)*x^k)/(1 - x). (End)

A344678 Coefficients for normal ordering of (x + D)^n and for the unsigned, probabilist's (or Chebyshev) Hermite polynomials H_n(x+y).

Original entry on oeis.org

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

Offset: 0

Tom Copeland, May 26 2021

This irregular triangular array contains the integer coefficients of the normal ordering of the expansions of the differential operator R = (x + D)^n with D = d/dx. This operator is the raising/creation operator for the unsigned, modified Hermite polynomials H_n(x) of A099174; i.e., R H_n(x) = H_{n+1}(x). The lowering/annihilation operator L is D; i.e, L H_n(x) = D H_n(x) = n H_{n-1}(x).
Generalizing to the ladder operators for S_n(x) a general Sheffer polynomial sequence, (L + R)^n 1 = H_n(S.(x)), where the umbral composition is defined by H_n(S.(x)) = (h. + S.(x))^n = Sum_{k = 0..n} binomial(n,k) h_{n-k} S_k(x), where h_n are the Taylor series coefficients of e^{t^2/2}; i.e., e^{h.t} = e^{t^2/2}.
Another interpretation is that the n-th row contains the coefficients of the terms in the normal ordering of the 2^n permutations of two binary symbols given by (L + R)^n under the Leibniz condition LR = RL + 1 equivalent to the commutator relation [L,R] = LR - RL = 1. For example, (x + D)^2 = xx + xD + Dx + DD = x^2 + 2 xD + 1 + D^2.
As in the examples, the coefficients are ordered as those for the terms x^k D^m, ordered first from higher k to lower k and subordinately from lower m to higher m.
The number sequences associated to these polynomials are intimately related to the complete graphs K_n, which have n vertices and (n-1) edges. H_n(x) are the independence polynomials for K_n. The moments, h_n, of the H_n(x) Appell polynomials are the aerated double factorials A001147, the number of perfect matchings in the complete graph K_{n}, zero for odd n. The row lengths, A002620, give the number of maximal strokes on the complete graph K_n. The triangular numbers--the sum of two consecutive row lengths--give the number of edges of K_n. The row sums, A005425, are the number of matchings of the corona K'_n of the complete graph K_n and the complete graph K_1. K_n can be viewed as the projection onto a plane of the edges of the regular (n-1)-dimensional simplex, whose face polynomials are (x+1)^n - 1 (cf. A135278 and A074909).
The coefficients represent the combinatorics of a switchboard problem in which among n subscribers, a subscriber may talk to one of the others, someone on an outside line, or not at all--no conference calls allowed. E.g., the coefficient 12 for H_4(x+y) is the number of ways among 4 subscribers that a pair of subscribers can talk to each other while another is on an outside line and the remaining subscriber is disconnected. The coefficient 3 for the polynomial corresponds to the number of ways two pairs of subscribers can talk among themselves. This can be regarded as a dinner table problem also where a diner may exchange positions with another diner; remain seated; or get up, change plans, and sit back down in the same chair--no more than one exchange per diner.
From Peter Luschny, May 28 2021: (Start)
If we write the monomials of the row polynomials in degree-lexicographic order, we observe: The coefficient triangle appears as a series of concatenated subtriangles. The first one is Pascal's triangle A007318. Appending the rows of triangle A094305 begins in row 2. In row 4, the next triangle starts, which is A344565. This scheme seems to go on indefinitely. [This is now set out in A344911.] (End)
From Tom Copeland, May 29 2021: (Start)
Simplex model: H_n(x+y) = (h. + x + y)^n with h_n the aerated odd double factorials (h_0 = 1, h_1 = 0, h_2 = 1, h_3 = 0, h_4 = 3, h_5 = 0, h_6 = 15, ...), the number of perfect matchings for the n vertices of the (n-1)-Dimensional simplices, i.e., the hypertriangles, or hypertetrahedrons. This multinomial enumerates permutations of three families of objects--vertices labeled with either x or y, or perfect (pair) matchings of the unlabeled vertices for the n vertices of the (n-1)-D simplex. For example, (x+D)^2 = xx + xD + Dx + D^2 = x^2 + 2xD + D^2 + 1 corresponds to H_2(x+y) = (h.+ x + y)^2 = h_0 (x+y)^2 + 2 h_1 (x+y) + h_2 = x^2 + 2xy + y^2 + 1, which, in turn, corresponds to a line segment, the 1-D simplex, with both vertices labeled with x's; or one with an x, the other a y; or both vertices labeled with y's; or one unlabeled matched pair. The exponent k of x^k y^m represents the number of these vertices to which an x is assigned, and m, those with a y assigned. The remaining unlabeled vertices (a sub-simplex) form an independent edge set of (single color) colored edges, i.e., no colored edge is touching another colored edge (a perfect matching for the sub-simplex) nor the vertices assigned an x or y. Accordingly, the coefficients for k=m=0 represent the number of ways to assign a perfect matching to the simplex--zero for simplices with an odd number of vertices and the odd double factorials (1, 3, 15, 105, ...) for the simplices with an even number. For the simplices with an odd number of vertices, the coefficients for k+m=1 are the odd double factorials. (Note the polynomials are invariant upon interchange of the variables x and y.) (End)

			(x + D)^0 = 1,
(x + D)^1 = x + D,
(x + D)^2 = x^2 + 2 x D + 1 + D^2,
(x + D)^3 = x^3 + 3 x^2 D + 3 x + 3 x D^2 + 3 D + D^3,
(x + D)^4 = x^4 + 4 x^3 D + 6 x^2 + 6 x^2 D^2 + 12 x D + 4 x D^3 + 3 + 6 D^2 + D^4.
(x + D)^5 = x^5 + 5 x^4 D + 10 x^3 + 10 x^3 D^2 + 30 x^2 D + 10 x^2 D^3 + 15 x + 30 x D^2 + 5 x D^4 + 15 D + 10 D^3 + D^5
H_6(x + y) = x^6 + 6 x^5 y + 15 x^4 + 15 x^4 y^2 + 60 x^3 y + 20 x^3 y^3 + 45 x^2 + 90 x^2 y^2 + 15 x^2 y^4 + 90 x y + 60 x y^3 + 6 x y^5 + 15 + 45 y^2 + 15 y^4 + y^6
H_7(x + y) = x^7 + 7 x^6 y + 21 x^5 + 21 x^5 y^2 + 105 x^4 y + 35 x^4 y^3 + 105 x^3 + 210 x^3 y^2 + 35 x^3 y^4 + 315 x^2 y + 210 x^2 y^3 + 21 x^2 y^5 + 105 x + 315 x y^2 + 105 x y^4 + 7 x y^6 + 105 y + 105 y^3 + 21 y^5 + y^7

P. Blasiak and P. Flajolet, Combinatorial Models of Creation-Annihilation, arXiv:1010.0354v3 [math.CO], 2011.
A. Varvak, Rook numbers and the normal ordering problem, arXiv:math/0402376v2 [math.CO], 2004

Cf. A000217, A001147, A002378, A002620, A005425, A074909, A099174, A135278.

Cf. A007318, A094305, A344565, A344911.

Last /@ CoefficientRules[#, {x, y}] & /@ Table[Simplify[(-y)^n (-2)^(-n/2) HermiteH[n, (x + 1/y)/Sqrt[-2]]], {n, 0, 7}] // Flatten (* Andrey Zabolotskiy, Mar 08 2024 *)

The bivariate e.g.f. is e^{t^2/2} e^{t(x + y)} = Sum_{n >= 0} H_n(x+y) t^n/n! = e^{t H.(x+y)} = e^{t (x + H.(y))}, as described below.
The coefficient of x^k D^m is n! h_{n-k-m} / [(n-k-m)! k! m!] with 0 <= k,m <= n and (k+m) <= n with h_n, as defined in the comments, aerated A001147.
Row lengths, r(n): 1, 2, 4, 6, 9, 12, 16, 20, 25, ... A002620(n).
Row sums: A005425 = 1, 2, 5, 14, 43, 142, ... .
The recursion H_{n+1}(x+y) = (x+y) H_n(x+y) + n H_{n-1}(x+y) follows from the differential raising and lowering operations of the Hermite polynomials.
The Baker-Campbell-Hausdorff-Dynkin expansion leads to the disentangling relation e^{t (x + D)} = e^{t^2/2} e^{tx} e^{tD} from which the formula above for the coefficients may be derived via differentiation with respect to t.
The row bivariate polynomials P_n(x,y) with y a commutative analog of D, or L, have the e.g.f. e^{-xy} e^{t(x + D)} e^{xy} = e^{-xy} e^{t^2/2} e^{tx} e{tD} e^{xy} = e^{t^2/2} e^{t(x + y)} = e^{t(h. + x + y)} = e^{t (x + H.(y))} = e^{t H.(x +y)}, so P_n(x,y) = H_n(x + y) = (x + H.(y))^n, the Hermite polynomials mentioned in the comments along with the umbral composition. The row sums are H_n(2), listed in A005425. For example, P_3(x,y) = (x + H.(y))^3 = x^3 H_0(y) + 3 x^2 H_1(y) + 3 x H_2(y) + H_3(y) = H_3(x+y) = (x+y)^3 + 3(x+y) = x^3 + 3 x^2 y + 3 x + 3 x y^2 + 3 y + y^2.
Alternatively, P_n(x,y) = H_n(x+y) = (z + d/dz)^n 1 with z replaced by (x+y) after the repeated differentiations since (x + D)^n 1 = H_n(x).
With initial index 1, the lengths r(n) of the rows of nonzero coefficients are the same as those for the polynomials given by 1 + (x+y)^2 + (x+y)^4 + ... + (x+y)^n for n even and for those for (x+y)^1 + (x+y)^3 + ... + (x+y)^n for n odd since the Hermite polynomials are even or odd polynomials. Consequently, r(n)= O(n) = 1 + 2 + 4 + ... n for n odd and r(n) = E(n) = 2 + 4 + ... + n for n even, so O(n) = ((n+1)/2)^2 and E(n) = (n/2)((n/2)+1) = n(n+2)/4 = 2 T(n/2) where T(k) are the triangular numbers defined by T(k) = 0 + 1 + 2 + 3 + ... + k = A000217(k). E(n) corresponds to A002378. Additionally, r(n) + r(n+1) = 1 + 2 + 3 + ... + n+1 = T(n+1).
From Tom Copeland, May 31 2021: (Start)
e^{D^2/2} = e^{h.D}, so e^{D^2/2} x^n = e^{h. D} x^n = (h. + x)^n = H_n(x) and e^{D^2/2} (x+y)^n = e^{h. D} (x+y)^n = (h. + x + y)^n = H_n(x+y).
From the Appell Sheffer polynomial calculus, the umbral compositional inverse of the sequence H_n(x+y), i.e., the sequence HI_n(x+y) such that H_n(HI.(x+y))  = (h. + HI.(x+y))^n = (h. + hi. + x + y)^n = (x+y)^n, is determined by e^{-t^2/2} = e^{hi. t}, so hi_n = -h_n and HI_n(x+y) = (-h. + x + y)^n = (-1)^n (h. - x - y)^n = (-1)^n H_n(-(x+y)). Then H_n(-H.(-(x+y))) = (x+y)^n.
In addition, HI_n(x) = (x - D) HI_{n-1}(x) = (x - D)^n 1 = e^{-D^2/2} x^n  = e^{hi. D} x^n = e^{-h. D} x^n with the e.g.f. e^{-t^2/2} e^{xt}.
The umbral compositional inversion property follows from x^n = e^{-D^2/2} e^{D^2/2} x^n = e^{-D^2/2} H_n(x) = H_n(HI.(x)) = e^{D^2/2} e^{-D^2/2} x^n = HI_n(H.(x)). (End)
The umbral relations above reveal that H_n(x+y) = (h. + x + y)^n = Sum_{k = 0..n} binomial(n,k) h_k (x+y)^{n-k}, which gives, e.g., for n = 3, H_3(x+y) = h_0 * (x+y)^3 + 3 h_1 * (x+y)^2 + 3 h_2 * (x+y) + h_3 = (x+y)^3 + 3 (x+y), the n-th through 0th rows of the Pascal matrix embedded within the n-th row of the Pascal matrix modulated by h_k. - Tom Copeland, Jun 01 2021
Varvak gives the coefficients of x^(n-m-k) D^{m-k} as n! / ( 2^k k! (n-k-m)! (m-k)! ), referring to them as the Weyl binomial coefficients, and derives them from rook numbers on Ferrers boards. (No mention of Hermite polynomials nor matchings on simplices are made.) Another combinatorial model and equivalent formula are presented in Blasiak and Flajolet (p. 16). References to much earlier work are given in both papers. - Tom Copeland, Jun 03 2021

A376826 Array read by antidiagonals: T(n,k) = n! * [x^n] exp(2x + (k/2)x^2), n >= 0, k >= 0.

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 1, 2, 5, 8, 1, 2, 6, 14, 16, 1, 2, 7, 20, 43, 32, 1, 2, 8, 26, 76, 142, 64, 1, 2, 9, 32, 115, 312, 499, 128, 1, 2, 10, 38, 160, 542, 1384, 1850, 256, 1, 2, 11, 44, 211, 832, 2809, 6512, 7193, 512, 1, 2, 12, 50, 268, 1182, 4864, 15374, 32400, 29186, 1024

Offset: 0

Andrew Howroyd, Oct 07 2024

			Array begins:
======================================================
n\k |   0    1    2     3     4     5     6      7 ...
----+-------------------------------------------------
  0 |   1    1    1     1     1     1     1      1 ...
  1 |   2    2    2     2     2     2     2      2 ...
  2 |   4    5    6     7     8     9    10     11 ...
  3 |   8   14   20    26    32    38    44     50 ...
  4 |  16   43   76   115   160   211   268    331 ...
  5 |  32  142  312   542   832  1182  1592   2062 ...
  6 |  64  499 1384  2809  4864  7639 11224  15709 ...
  7 | 128 1850 6512 15374 29696 50738 79760 118022 ...
     ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals)
Arvind Ayyer, Hiranya Kishore Dey and Digjoy Paul, How large is the character degree sum compared to the character table sum for a finite group?, arXiv preprint arXiv:2406.06036, [math.RT], 2024.

Columns 0..5 are A000079, A005425, A000898, A202830, A193778, A202832.

Cf. A359762, A373625.

T(n,k) = {sum(i=0, n\2, binomial(n,2*i) * 2^(n-2*i) * k^i * (2*i)!/(2^i*i!))}

E.g.f. of column k: exp(2*x + k*x^2/2).
Column k is the binomial transform of column k of A359762.
T(n,k) = Sum_{i=0..floor(n/2)} binomial(n,2*i) * 2^(n-2*i) * k^i * (2*i-1)!!.
T(n,k) = Sum_{i=0..floor(n/2)} 2^(n-3*i) * k^i * n! / ((n-2*i)! * i!).

A093620 Values of Laguerre polynomials: a(n) = 2^nn!LaguerreL(n,-1/2,-2).

Original entry on oeis.org

1, 5, 43, 499, 7193, 123109, 2430355, 54229907, 1347262321, 36833528197, 1097912385851, 35409316648435, 1227820993510153, 45528569866101989, 1797044836586213923, 75200136212985945619, 3324579846014080352225, 154797474251689486249477, 7570037033145534341015371

Offset: 0

Karol A. Penson, Apr 06 2004

nonn

Not the same as the numerator of LaguerreL(n,-1/2,-2). - Robert G. Wilson v, Apr 08 2004

Alois P. Heinz, Table of n, a(n) for n = 0..200

Bisection of A005425.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(4*x/(1-2*x))/(1-2*x)^(1/2))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 11 2018

a:= proc(n) option remember; `if`(n<2, 4*n+1,
      (4*n+1)*a(n-1) -2*(n-1)*(2*n-3)*a(n-2))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Jun 22 2013

Table[2^n n!LaguerreL[n, -1/2, -2], {n, 0, 16}] (* Robert G. Wilson v, Apr 08 2004 *)

x='x+O('x^30); Vec(serlaplace(exp(4*x/(1-2*x))/(1-2*x)^(1/2))) \\ G. C. Greubel, May 11 2018

a(n) = 2^n*n!*pollaguerre(n, -1/2, -2); \\ Michel Marcus, Feb 05 2021

E.g.f.: exp(4*x/(1-2*x))/(1-2*x)^(1/2).

a(n) ~ n^n*2^(n-1/2)*exp(-n+2*sqrt(2*n)-1) * (1 + 5/(6*sqrt(2*n))). - Vaclav Kotesovec, Jun 22 2013

More terms from Robert G. Wilson v, Apr 08 2004

A094073 Coefficients arising in combinatorial field theory.

Original entry on oeis.org

4, 240, 49938, 24608160, 23465221750, 38341895571708, 98780305524248572, 377796303580335320432, 2048907276496726375662702, 15198414983297581845761672560, 149768511689247547252666676150490

Offset: 1

N. J. A. Sloane, May 01 2004

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..160
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"
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).

Cf. A000085, A005425, A094070, A094071, A094072, A094074.

Cf. A000110.

with(combinat): a:=n->bell(2*n)*(2*n)!*coeff(series(exp(x*sinh(x)), x=0,40), x^(2*n)): seq(a(n),n=1..13); # Emeric Deutsch, Jan 22 2005

a[n_] := (2n)! BellB[2n] SeriesCoefficient[Exp[x Sinh[x]], {x, 0, 2n}];
Table[a[n], {n, 1, 11}] (* Jean-François Alcover, Nov 11 2018 *)

a(n) = (2n)!*bell(2n)*coeff(exp(x*sinh(x)), x^(2n)). - Emeric Deutsch, Jan 22 2005

More terms from Emeric Deutsch, Jan 22 2005

A094074 Coefficients arising in combinatorial field theory.

Original entry on oeis.org

1, 5, 129, 7485, 755265, 116338005, 25263540225, 7328358482445, 2730934406225025, 1269262202389906725, 718835160819268317825, 486853691847850902700125, 388278919916351519293663425

Offset: 0

N. J. A. Sloane, May 01 2004

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..225
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"
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).
A. Horzela, P. Blasiak, G. E. H. Duchamp, K. A. Penson and A. I. Solomon, A product formula and combinatorial field theory, arXiv:quant-ph/0409152, 2004.

Equals A001147(n) * A093620(n).

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

a[n_] := (2n)! SeriesCoefficient[(1-x^2)^(-1/2) Exp[2x^2/(1-x^2)], {x, 0, 2n}];
Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Nov 11 2018 *)

a(n) = (2n)!/(2^n*n!) * h(2n, 2), with h(n, x) the polynomials in A099174.

E.g.f.: Sum_{n>=0} a(n)*x^(2n)/(2n)! = (1-x^2)^(-1/2) * exp(2x^2/(1-x^2)).

Edited and extended by Ralf Stephan, Oct 14 2004.

A100862 Triangle read by rows: T(n,k) is the number of k-matchings of the corona K'(n) of the complete graph K(n) and the complete graph K(1); in other words, K'(n) is the graph constructed from K(n) by adding for each vertex v a new vertex v' and the edge vv'.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 6, 6, 1, 1, 10, 21, 10, 1, 1, 15, 55, 55, 15, 1, 1, 21, 120, 215, 120, 21, 1, 1, 28, 231, 665, 665, 231, 28, 1, 1, 36, 406, 1736, 2835, 1736, 406, 36, 1, 1, 45, 666, 3990, 9891, 9891, 3990, 666, 45, 1, 1, 55, 1035, 8310, 29505, 45297, 29505, 8310, 1035, 55, 1, 1, 66, 1540, 16005, 77715, 173712, 173712, 77715, 16005, 1540, 66, 1

Offset: 0

Emeric Deutsch, Jan 08 2005

Row n has n+1 terms.

Row sums yield A005425.

			T(3,2)=6 because in the graph with vertex set {A,B,C,a,b,c} and edge set {AB,AC,BC,Aa,Bb,Cc} we have the following six 2-matchings: {Aa,BC},{Bb,AC},{Cc,AB},{Aa,Bb},{Aa,Cc} and {Bb,Cc}.
Triangle starts:
  1;
  1,  1;
  1,  3,  1;
  1,  6,  6,  1;
  1, 10, 21, 10,  1;
  1, 15, 55, 55, 15,  1;

Paul Barry, On a Generalization of the Narayana Triangle, J. Int. Seq., Vol. 14 (2011), Article 11.4.5.
Paul Barry, Constructing Exponential Riordan Arrays from Their A and Z Sequences, Journal of Integer Sequences, Vol. 17 (2014), Article 14.2.6.
Paul Barry, The Gamma-Vectors of Pascal-like Triangles Defined by Riordan Arrays, arXiv:1804.05027 [math.CO], 2018.

Cf. A005425.

Cf. A107102 (matrix inverse).

P[0]:=1: for n from 1 to 11 do P[n]:=sort(expand((1+t)*P[n-1]+(n-1)*t*P[n-2])) od: for n from 0 to 11 do seq(coeff(t*P[n],t^k),k=1..n+1) od; # gives the sequence in triangular form

P[0] = 1; P[1] = 1+t; P[n_] := P[n] = (1+t) P[n-1] + (n-1) t P[n-2];
Table[CoefficientList[P[n], t], {n, 0, 11}] // Flatten (* Jean-François Alcover, Jul 23 2018 *)

{T(n,k)=local(X=x+x*O(x^n),Y=y+y*O(y^k)); n!*polcoeff(polcoeff(exp(X+Y*X^2/2+X*Y),n,x),k,y)} \\ Paul D. Hanna, Jul 18 2005

T(n,k) = T(n,n-k).
E.g.f.: exp(z+t*z+t*z^2/2).
Row generating polynomial P[n] = [ -i*sqrt(t/2)]^n*H(n, i(1+t)/sqrt(2t)), where H(n, x) is a Hermite polynomial and i=sqrt(-1).
Row generating polynomials P[n] satisfy P[0]=1, P[n]=(1+t)P[n-1]+(n-1)tP[n-2].
From Fabián Pereyra, Jan 05 2022: (Start)
T(n,k) = T(n-1,k) + (n-1)*T(n-2,k-1) + T(n-1,k-1), n>=0, 0<=k<=n; T(0,0) = 1.
T(n,k) = Sum_{j=k..n} C(n,j)*B(j,j-k), where B are the Bessel numbers A100861. (End)

A111062 Triangle T(n, k) = binomial(n, k) * A000085(n-k), 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 4, 6, 3, 1, 10, 16, 12, 4, 1, 26, 50, 40, 20, 5, 1, 76, 156, 150, 80, 30, 6, 1, 232, 532, 546, 350, 140, 42, 7, 1, 764, 1856, 2128, 1456, 700, 224, 56, 8, 1, 2620, 6876, 8352, 6384, 3276, 1260, 336, 72, 9, 1, 9496, 26200, 34380, 27840, 15960, 6552, 2100, 480, 90, 10, 1

Offset: 0

Philippe Deléham, Oct 07 2005

Triangle related to A000085.
Riordan array [exp(x(2+x)/2),x]. - Paul Barry, Nov 05 2008
Array is exp(S+S^2/2) where S is A132440 the infinitesimal generator for Pascal's triangle. T(n,k) gives the number of ways to choose a subset of {1,2,...,n} of size k and then partitioning the remaining n-k elements into sets each of size 1 or 2. Cf. A122832. - Peter Bala, May 14 2012
T(n,k) is equal to the number of R-classes (equivalently, L-classes) in the D-class consisting of all rank k elements of the partial Brauer monoid of degree n. - James East, Aug 17 2015

			Rows begin:
     1;
     1,    1;
     2,    2,    1;
     4,    6,    3,    1;
    10,   16,   12,    4,    1;
    26,   50,   40,   20,    5,    1;
    76,  156,  150,   80,   30,    6,   1;
   232,  532,  546,  350,  140,   42,   7,  1;
   764, 1856, 2128, 1456,  700,  224,  56,  8, 1;
  2620, 6876, 8352, 6384, 3276, 1260, 336, 72, 9, 1;
From _Paul Barry_, Apr 23 2009: (Start)
Production matrix is:
  1, 1,
  1, 1, 1,
  0, 2, 1, 1,
  0, 0, 3, 1, 1,
  0, 0, 0, 4, 1, 1,
  0, 0, 0, 0, 5, 1, 1,
  0, 0, 0, 0, 0, 6, 1, 1,
  0, 0, 0, 0, 0, 0, 7, 1, 1,
  0, 0, 0, 0, 0, 0, 0, 8, 1, 1 (End)
From _Peter Bala_, Feb 12 2017: (Start)
The infinitesimal generator has integer entries and begins
  0
  1  0
  1  2  0
  0  3  3  0
  0  0  6  4  0
  0  0  0 10  5  0
  0  0  0  0 15  6  0
  ...
and is the generalized exponential Riordan array [x + x^2/2!,x].(End)

Muniru A Asiru, Table of n, a(n) for n = 0..1325
Igor Dolinka, James East, Athanasios Evangelou, Des FitzGerald, Nicholas Ham, James Hyde, and Nicholas Loughlin, Enumeration of idempotents in diagram semigroups and algebras, arXiv:1408.2021 [math.GR], 2014.
Igor Dolinka, James East, Athanasios Evangelou, Des FitzGerald, Nicholas Ham, James Hyde, and Nicholas Loughlin, Enumeration of idempotents in diagram semigroups and algebras, J. Combin. Theory Ser. A 131 (2015), 119-152.
Tom Halverson and Theodore N. Jacobson, Set-partition tableaux and representations of diagram algebras, arXiv:1808.08118 [math.RT], 2018.

Cf. A000085, A005425 (row sums), A007318, A013989, A122832, A132440.

Cf. A099174, A133314, A159834 (inverse matrix).

Flat(List([0..10],n->List([0..n],k->(Factorial(n)/Factorial(k))*Sum([0..n-k],j->Binomial(j,n-k-j)/(Factorial(j)*2^(n-k-j)))))); # Muniru A Asiru, Jun 29 2018

a[n_] := Sum[(2 k - 1)!! Binomial[n, 2 k], {k, 0, n/2}]; Table[Binomial[n, k] a[n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Aug 20 2015, after Michael Somos at A000085 *)

def A111062_triangle(dim):
    M = matrix(ZZ,dim,dim)
    for n in (0..dim-1): M[n,n] = 1
    for n in (1..dim-1):
        for k in (0..n-1):
            M[n,k] = M[n-1,k-1]+M[n-1,k]+(k+1)*M[n-1,k+1]
    return M
A111062_triangle(9) # Peter Luschny, Sep 19 2012

Sum_{k>=0} T(m, k)*T(n, k)*k! = T(m+n, 0) = A000085(m+n).
Sum_{k=0..n} T(n, k) = A005425(n).
Apparently satisfies T(n,m) = T(n-1,m-1) + T(n-1,m) + (m+1) * T(n-1,m+1). - Franklin T. Adams-Watters, Dec 22 2005 [corrected by Werner Schulte, Feb 12 2025]
T(n,k) = (n!/k!)*Sum_{j=0..n-k} C(j,n-k-j)/(j!*2^(n-k-j)). - Paul Barry, Nov 05 2008
G.f.: 1/(1-xy-x-x^2/(1-xy-x-2x^2/(1-xy-x-3x^2/(1-xy-x-4x^2/(1-... (continued fraction). - Paul Barry, Apr 23 2009
T(n,k) = C(n,k)*Sum_{j=0..n-k} C(n-k,j)*(n-k-j-1)!! where m!!=0 if m is even. - James East, Aug 17 2015
From Tom Copeland, Jun 26 2018: (Start)
E.g.f.: exp[t*p.(x)] = exp[t + t^2/2] e^(x*t).
These polynomials (p.(x))^n = p_n(x) are an Appell sequence with the lowering and raising operators L = D and R = x + 1 + D, with D = d/dx, such that L p_n(x) = n * p_(n-1)(x) and R p_n(x) = p_(n+1)(x), so the formalism of A133314 applies here, giving recursion relations.
The transpose of the production matrix gives a matrix representation of the raising operator R.
exp(D + D^2/2) x^n= e^(D^2/2) (1+x)^n = h_n(1+x) = p_n(x) = (a. + x)^n, with (a.)^n = a_n = A000085(n) and h_n(x) the modified Hermite polynomials of A099174.
A159834 with the e.g.f. exp[-(t + t^2/2)] e^(x*t) gives the matrix inverse for this entry with the umbral inverse polynomials q_n(x), an Appell sequence with the raising operator  x - 1 - D, such that umbrally composed q_n(p.(x)) = x^n = p_n(q.(x)). (End)

Corrected by Franklin T. Adams-Watters, Dec 22 2005

10th row added by James East, Aug 17 2015

A085483 Triangle read by rows: S_B(n,k) = "Type B" Stirling numbers of the second kind.

Original entry on oeis.org

2, 2, 5, 2, 15, 14, 2, 35, 84, 43, 2, 75, 350, 430, 142, 2, 155, 1260, 2795, 2130, 499, 2, 315, 4214, 15050, 19880, 10479, 1850, 2, 635, 13524, 73143, 149100, 132734, 51800, 7193, 2, 1275, 42350, 334110, 987042, 1320354, 854700, 258948, 29186, 2, 2555, 130620, 1466515, 6038550, 11390673, 10878000, 5394750, 1313370, 123109

Offset: 1

James East, Aug 15 2003

			S_B(2,2)=5 because the relevant partitions of {-2,-1,1,2} are: {-2|-1|1|2}, {-1,1|-2|2}, {-1|1|-2,2}, {-1,1|-2,2}, {1,-2|-1,2}.
Triangle begins:
  2;
  2,   5;
  2,  15,   14;
  2,  35,   84,   43;
  2,  75,  350,  430,  142;
  2, 155, 1260, 2795, 2130, 499;
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows n = 1..150, flattened)
Eli Bagno and David Garber, Combinatorics of q,r-analogues of Stirling numbers of type B, arXiv:2401.08365 [math.CO], 2024. See page 4.
Takao Komatsu, Eli Bagno, and David Garber, A q,r-analogue of poly-Stirling numbers of second kind with combinatorial applications, arXiv:2209.06674 [math.CO], 2022.

Cf. A008277, A005425.

S_B(n, 1) + ... + S_B(n, n) = A002872(n).

nn = 10; f[n_] := Sum[2^(n - 3 k) n!/((n - 2 k)! k!), {k, 0, n}]; Do[f[n], {n, 0, nn}]; Table[f[k]*StirlingS2[n, k], {n, nn}, {k, n}] (* Michael De Vlieger, Sep 21 2022, after Robert G. Wilson v at A005425 *)

A partition of {-n, ..., -1, 1, ..., n} into nonempty subsets X_1, ..., X_r is called "symmetric" if for each i -X_i = X_j for some j. S_B(n, k) is the number of such symmetric partitions whose induced partition on {1, ..., n} involves k nonempty subsets. S_B(n, k) = S(n, k) * a(k), where S(n, k) is A008277 and a(k) is A005425.

A094071 Coefficients arising in combinatorial field theory.

Original entry on oeis.org

1, 2, 10, 75, 572, 6293, 92962, 1395180, 25482135, 582310475, 13697614020, 364311810217, 11551145067139, 380339218683310, 13636394439014770, 563142483841155427, 24264229405883569164, 1114389674994185476663

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, 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, A094070, A094072, A094073, A094074.

Cf. A000110.

with(combinat):F:=series(exp(x+x^3/3!),x=0,25): seq((n+1)!*coeff(F,x^(n+1))*bell(n+1),n=0..20);

a[n_] := (n+1)! BellB[n+1] SeriesCoefficient[Exp[x+x^3/3!], {x, 0, n+1}];
Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Nov 11 2018 *)

a(n)=(n+1)!*B(n+1)*[x^(n+1)](exp(x+x^3/3!)), where B(n) are the Bell numbers (A000110) - Emeric Deutsch, Nov 23 2004

More terms from Emeric Deutsch, Nov 23 2004

cp's OEIS Frontend

A128227 Right border (1,1,1,...) added to A002260.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Maxima

Python

Python

Formula

A344678 Coefficients for normal ordering of (x + D)^n and for the unsigned, probabilist's (or Chebyshev) Hermite polynomials H_n(x+y).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A376826 Array read by antidiagonals: T(n,k) = n! * [x^n] exp(2*x + (k/2)*x^2), n >= 0, k >= 0.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A093620 Values of Laguerre polynomials: a(n) = 2^n*n!*LaguerreL(n,-1/2,-2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Formula

Extensions

A094073 Coefficients arising in combinatorial field theory.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A094074 Coefficients arising in combinatorial field theory.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A100862 Triangle read by rows: T(n,k) is the number of k-matchings of the corona K'(n) of the complete graph K(n) and the complete graph K(1); in other words, K'(n) is the graph constructed from K(n) by adding for each vertex v a new vertex v' and the edge vv'.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A376826 Array read by antidiagonals: T(n,k) = n! * [x^n] exp(2x + (k/2)x^2), n >= 0, k >= 0.

A093620 Values of Laguerre polynomials: a(n) = 2^nn!LaguerreL(n,-1/2,-2).