A323841 -id:A323841 - cp's OEIS frontend

A007889 Number of intransitive (or alternating, or Stanley) trees: vertices are [0,n] and for no i

1, 1, 2, 7, 36, 246, 2104, 21652, 260720, 3598120, 56010096, 971055240, 18558391936, 387665694976, 8787898861568, 214868401724416, 5636819806209792, 157935254554567296, 4707152127520549120, 148704074888134683520, 4963548160096887021056, 174553183413968718996736

Offset: 0

Alexander Postnikov [ apost(AT)math.mit.edu ]

Number of local binary search trees (i.e. labeled binary trees such that every left child has a smaller label than its parent and every right child has a larger label than its parent) on n vertices. Example: a(3)=7 because we have 3L2L1, 2L1R3, 3L1R2, 1R2R3, 1R3L2, 2R3L1 (Li means left child labeled i, RI means right child labeled i) and root 2 with left child 1 and right child 3. - Emeric Deutsch, Nov 24 2004

Number of regions of the Linial arrangement. - Ira M. Gessel, Nov 01 2023

I. M. Gelfand, M. I. Graev and A. Postnikov, Combinatorics of hypergeometric functions associated with positive roots, in Arnold-Gelfand Mathematical Seminars: Geometry and Singularity Theory, Birkhäuser, 1997.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.41(a).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
C. Chauve, S. Dulucq and A. Rechnitzer, Enumerating alternating trees, J. Combin. Theory Ser. A 94 (2001), 142-151.
Sergey Fomin and Grigory Mikhalkin, Labeled floor diagrams for plane curves, arXiv:0906.3828, 2009-2010. [_N. J. A. Sloane_, Sep 27 2010]
G. Hetyei, Efron's coins and the Linial arrangement, arXiv preprint arXiv:1511.04482 [math.CO], 2015.
D. E. Knuth, Letter to Daniel Ullman and others, Apr 29 1997 [Annotated scanned copy, with permission]
A. Postnikov, Intransitive Trees, J. Combin. Theory Ser. A 79 (1997), 360-366.
Richard P. Stanley, Hyperplane arrangements, interval orders, and trees, Proc. Natl. Acad. Sci. USA 93 (1996), 2620-2625.
Index entries for sequences related to trees

Cf. A038049, A138860, A283828, A323841.

Row sums of A029847.

f:= n->1/(2^n*(n+1))*add(binomial(n+1, k)*k^n, k=1..(n+1)): seq(f(n), n=0..19);

With[{nn=20},CoefficientList[Series[-2/x LambertW[-1/2x Exp[x/2]], {x,0,nn}], x]Range[0,nn]!] (* Harvey P. Dale, Aug 12 2011 *)
Table[1/((n+1)2^n) Sum[Binomial[n+1,k]k^n,{k,n+1}],{n,0,20}] (* Harvey P. Dale, Apr 21 2012 *)

{a(n)=local(A=1+x);for(i=0,n,A=exp(x*(1+A)/2 +x*O(x^n)));n!*polcoeff(A,n)} \\ Paul D. Hanna, Mar 29 2008

/* Coefficients of A(x)^p are given by: */ {a(n,p=1)=(1/2^n)*sum(k=0,n,binomial(n,k)*p*(k+p)^(n-1))} \\ Vladeta Jovovic and Paul D. Hanna, Apr 03 2008

def A007889(n) : return add(binomial(n,k)*(k+1)^(n-1) for k in (0..n))/2^n
for n in (0..19) : print(A007889(n)) # Peter Luschny, Feb 29 2012

a(n) = (1/((n+1)*2^n))*Sum_{k=1..n+1} C(n+1,k)*k^n.
E.g.f. A(x) satisfies: A(x) = exp( x*(1 + A(x))/2 ). E.g.f. A(x) equals the inverse function of 2*log(x)/(1+x). - Paul D. Hanna, Mar 29 2008
E.g.f.: -2/x*LambertW(-1/2*x*exp(1/2*x)). - Vladeta Jovovic, Mar 29 2008
From Vladeta Jovovic and Paul D. Hanna, Apr 03 2008: (Start)
Powers of e.g.f.: If A(x)^p = Sum_{n>=0} a(n,p)*x^n/n! then a(n,p) = (1/2^n)* Sum_{k=0..n} binomial(n,k)*p*(k+p)^(n-1).
Let A(x) = e.g.f. of A007889, B(x) = e.g.f. of A138860 where B(x) = exp( x*[B(x) + B(x)^2]/2 ); then B(x) = A(x*B(x)) = (1/x)*Series_Reversion(x/A(x)) and A(x) = B(x/A(x)) = x/Series_Reversion(x*B(x)). (End)
For n>=2, a(n)=Sum_{1,...,floor(n/2)}binomial(n-1, 2k-1)*k^(n-2). [Vladimir Shevelev, Mar 21 2010]
For n>0, a(n) = A088789(n+1)*2/(n+1). [Vaclav Kotesovec, Dec 26 2011]

A135922 Inverse binomial transform of A006116, which is the sum of Gaussian binomial coefficients [n,k] for q=2.

Original entry on oeis.org

1, 1, 2, 6, 26, 158, 1330, 15414, 245578, 5382862, 162700898, 6801417318, 394502066810, 31849226170622, 3589334331706258, 566102993389615254, 125225331231990004138, 38920655753545108286254, 17021548688670112527781058, 10486973328106497739526535366

Offset: 0

Paul D. Hanna, Dec 06 2007

nonn

Let B = {v_1,v_2,...,v_n} be a basis for F_2^n.  a(n) is the number of subspaces of F_2^n that do not contain any of the vectors in B.  Moreover, for 1<=k<=n, let S be a size k subset of B. a(k) is the number of subspaces of F_2^n that do not contain any of the vectors in S but do contain all the vectors in B - S. - Geoffrey Critzer, May 03 2025
Also number of Stanley graphs on n nodes. For precise definition see Knuth (1997). - Alois P. Heinz, Sep 24 2019
Also the number of naturally labeled posets on [n] with height at most two. - David Bevan, Jul 28 2022; Nov 16 2023
Also the number of sign mappings X:([n] choose 2) -> {+,-} such that for any ordered 3-tuple aManfred Scheucher, Jan 05 2024

			O.g.f.: A(x) = 1 + x/(1-x) + x^2/((1-x)*(1-3x)) + x^3/((1-x)*(1-3x)*(1-7x)) + x^4/((1-x)*(1-3x)*(1-7x)*(1-15x)) + ...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 318.

Alois P. Heinz, Table of n, a(n) for n = 0..115
David Bevan, Gi-Sang Cheon and Sergey Kitaev, On naturally labelled posets and permutations avoiding 12-34, arXiv:2311.08023 [math.CO], 2023.
Lucas Gagnon, The combinatorics of normal subgroups in the unipotent upper triangular group, arXiv:2012.00108 [math.CO], 2020.
D. E. Knuth, Letter to Daniel Ullman and others, Apr 29 1997 [Annotated scanned copy, with permission]
Zvi Rosen and Yan X. Zhang, Convex Neural Codes in Dimension 1, arXiv:1702.06907 [math.CO], 2017. Mentions this sequence.
R. P. Stanley, Problem 10572, The American Mathematical Monthly, 104(2) (1997), 168.
R. P. Stanley and S. C. Locke, Graphs without increasing paths: Solution to Problem 10572, The American Mathematical Monthly, 106(2) (1999), 168.

Cf. A006116, A323841, A323842, A323843, A139382, A006455, A356111.

b:= proc(n) option remember; add(mul(
      (2^(i+k)-1)/(2^i-1), i=1..n-k), k=0..n)
    end:
a:= proc(n) option remember;
      add(b(n-j)*binomial(n,j)*(-1)^j, j=0..n)
    end:
seq(a(n), n=0..19);  # Alois P. Heinz, Sep 24 2019

Table[SeriesCoefficient[Sum[x^n/Product[(1-(2^k-1)*x),{k,0,n}],{n,0,nn}],{x,0,nn}] ,{nn,0,20}] (* Vaclav Kotesovec, Aug 23 2013 *)
b[n_] := b[n] = Sum[Product[(2^(i+k)-1)/(2^i-1), {i, 1, n-k}], {k, 0, n}];
a[n_] := a[n] = Sum[(-1)^j b[n-j] Binomial[n, j], {j, 0, n}];
a /@ Range[0, 19] (* Jean-François Alcover, Mar 10 2020, after Alois P. Heinz *)

a(n)=polcoeff(sum(k=0, n, x^k/prod(j=0, k, 1-(2^j-1)*x+x*O(x^n))), n)

# After Vladimir Kruchinin.
def a(n):
    @cached_function
    def T(n, k):
        if k == 1 or k == n: return 1
        return (2^k-1)*T(n-1, k) + T(n-1, k-1)
    return sum(T(n, k) for k in (1..n)) if n > 0 else 1
print([a(n) for n in (0..19)]) # Peter Luschny, Feb 26 2020

O.g.f.: A(x) = Sum_{n>=0} x^n / Product_{k=0..n} (1 - (2^k-1)*x).
G.f.: (G(0) - 1)/(x-1) where G(k) = 1 - 1/(1-x*(2^k-1))/(1-x/(x-1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jan 16 2013
a(n) ~ c * 2^(n^2/4), where c = EllipticTheta[3,0,1/2]/QPochhammer[1/2,1/2] = 7.3719688014613... if n is even and c = EllipticTheta[2,0,1/2]/QPochhammer[1/2,1/2] = 7.3719494907662... if n is odd. - Vaclav Kotesovec, Aug 23 2013
a(n) = Sum_{k=0..n} qStirling2(n,k), where qStirling2 is the triangle A139382. - Vladimir Kruchinin, Feb 26 2020
G.f.: f(1), where f(y) = 1 + x*((y-1)*f(y) + f(2*y)). - David Bevan, Jul 28 2022
E.g.f.:  exp(-x)*g(x) where g(x) is the e.g.f. for A006116. (given in D. E. Knuth link) - Geoffrey Critzer, May 03 2025

References for Stanley graphs added by David Bevan, Jul 24 2024

A323842 Number of n-node Stanley graphs without isolated nodes.

Original entry on oeis.org

1, 0, 1, 2, 11, 72, 677, 8686, 152191, 3632916, 118317913, 5271781946, 322549557299, 27208234437984, 3177021912874253, 515436469519284358, 116581257420399219175, 36866447823471507563436, 16339685138335030408029889, 10170100145132835334268145362

Offset: 0

N. J. A. Sloane, Feb 04 2019

nonn

For precise definition see Knuth (1997).

Also, the number of naturally labeled posets on [n] with height at most two and no isolated elements. - David Bevan, Nov 17 2023

Alois P. Heinz, Table of n, a(n) for n = 0..115
David Bevan, Gi-Sang Cheon and Sergey Kitaev, On naturally labelled posets and permutations avoiding 12-34, arXiv:2311.08023 [math.CO], 2023.
D. E. Knuth, Letter to Daniel Ullman and others, Apr 29 1997. [Annotated scanned copy, with permission]

Cf. A135922, A323841, A323843, A323502, A356111.

b:= proc(n) option remember; add(mul(
      (2^(i+k)-1)/(2^i-1), i=1..n-k), k=0..n)
    end:
g:= proc(n) option remember;
      add(b(n-j)*binomial(n, j)*(-1)^j, j=0..n)
    end:
a:= proc(n) option remember;
      add(g(n-j)*binomial(n, j)*(-1)^j, j=0..n)
    end:
seq(a(n), n=0..21);  # Alois P. Heinz, Sep 24 2019

b[n_] := b[n] = Sum[Product[(2^(i+k) - 1)/(2^i - 1), {i, n-k}], {k, 0, n}];
g[n_] := g[n] = Sum[b[n-j] Binomial[n, j] (-1)^j, {j, 0, n}];
a[n_] := a[n] = Sum[g[n-j] Binomial[n, j] (-1)^j, {j, 0, n}];
a /@ Range[0, 21] (* Jean-François Alcover, May 24 2020, after Alois P. Heinz *)

P(n, k, x):=if k<0 or n<0 then 0 else if k=0 then 1 else x*P(n, k-1, x)+
2^k*P(n-1, k, (x+1)/2);
a(n):=sum(P(n-k, k, -1), k, 0, n);
makelist(a(n), n, 0, 10);
/* Vladimir Kruchinin, Mar 08 2020 */

a(n) = Sum_{j=0..n} (-1)^j * C(n,j) * A135922(n-j). - Alois P. Heinz, Sep 24 2019
a(n) = Sum_{k=0..n} P(n-k, k, -1), where  P(n, k, x) = x*P(n, k-1, x) + 2^k*P(n-1, k, (x+1)/2). - Vladimir Kruchinin, Mar 09 2020
G.f.: g(1,0), where g(u,v) = 1 + x*((v-1)*g(u,v) + g(2*u,u+v)). - David Bevan, Jul 28 2022
G.f.: 1/(1+z) * Sum_{k>=0} (z^k / Prod_{i=2..k} (1 - (2^i-2)*z)). - David Bevan, Nov 17 2023; simplified on Jul 25 2024

More terms from Alois P. Heinz, Sep 24 2019

A323843 Number of n-node connected Stanley graphs.

Original entry on oeis.org

0, 1, 1, 2, 8, 52, 502, 6824, 127166, 3205924, 108975934, 5006366048, 312601245662, 26708244267148, 3142852107059758, 512229404374936616, 116165284523764481294, 36791597841822774872116, 16320947226945992981680606, 10163558457757761048966068912

Offset: 0

N. J. A. Sloane, Feb 04 2019

nonn

For precise definition see Knuth (1997).

Alois P. Heinz, Table of n, a(n) for n = 0..155
D. E. Knuth, Letter to Daniel Ullman and others, Apr 29 1997. [Annotated scanned copy, with permission]

Cf. A135922, A323841, A323842.

b:= proc(n) option remember; add(mul(
      (2^(i+k)-1)/(2^i-1), i=1..n-k), k=0..n)
    end:
p:= proc(n) option remember;
      add(b(n-j)*binomial(n, j)*(-1)^j, j=0..n)
    end:
a:= proc(n) option remember; `if`(n=0, 0, p(n)-add(
      binomial(n, j)*p(n-j)*a(j)*j, j=1..n-1)/n)
    end:
seq(a(n), n=0..21);  # Alois P. Heinz, Sep 24 2019

b[n_] := b[n] = Sum[Product[(2^(i+k) - 1)/(2^i - 1), {i, n-k}], {k, 0, n}];
p[n_] := p[n] = Sum[b[n-j] Binomial[n, j] (-1)^j, {j, 0, n}];
a[n_] := a[n] = If[n == 0, 0, p[n] - Sum[Binomial[n, j] p[n-j] a[j] j, {j, n-1}]/n];
a /@ Range[0, 21] (* Jean-François Alcover, May 24 2020, after Alois P. Heinz *)

More terms from Alois P. Heinz, Sep 24 2019

A029847 Gessel-Stanley triangle read by rows: triangle of coefficients of polynomials arising in connection with enumeration of intransitive trees by number of nodes and number of right nodes.

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 1, 1, 17, 17, 1, 1, 49, 146, 49, 1, 1, 129, 922, 922, 129, 1, 1, 321, 4887, 11234, 4887, 321, 1, 1, 769, 23151, 106439, 106439, 23151, 769, 1, 1, 1793, 101488, 856031, 1679494, 856031, 101488, 1793, 1, 1, 4097, 420512, 6137832, 21442606, 21442606, 6137832, 420512, 4097, 1

Offset: 0

N. J. A. Sloane

For precise definition see Knuth (1997).

Named after the American mathematicians Ira Martin Gessel (b. 1951) and Richard Peter Stanley (b. 1944). - Amiram Eldar, Jun 11 2021

			Triangle begins:
  1;
  .   1;
  .   1,   1;
  .   1,   5,   1;
  .   1,  17,  17,   1;
  .   1,  49, 146,  49,   1;
  .   1, 129, 922, 922, 129, 1;
  .   ...

Alois P. Heinz, Rows n = 0..141, flattened
Donald E. Knuth, Letter to Daniel Ullman and others, Apr 29 1997. [Annotated scanned copy, with permission]
Alexander Postnikov, Intransitive Trees, J. Combin. Theory Ser. A, Vol. 79, No. 2 (1997), pp. 360-366.

Row sums give A007889.

f:= proc(n,k) option remember; `if`(k<0, 0, `if`(n=0
      and k=0, 1, f(n-1,k-1)+add(add(binomial(n-1, l)
      *s*f(l,s)*f(n-l-1,k-s), s=1..l), l=1..n-1)))
    end:
seq(seq(f(n, k), k=min(n, 1)..n), n=0..10); # Alois P. Heinz, Sep 24 2019

f[n_, k_] := f[n, k] = If[k<0, 0, If[n==0 && k==0, 1, f[n-1, k-1]+Sum[Sum[ Binomial[n-1, l]*s*f[l, s]*f[n-l-1, k-s], {s, 1, l}], {l, 1, n-1}]]];
Table[Table[f[n, k], {k, Min[n, 1], n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 14 2021, after Alois P. Heinz *)

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 23 2003

A329154 Coefficients of polynomials related to the sum of Gaussian binomial coefficients for q = 2. Triangle read by rows, T(n,k) for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 6, 6, 3, 1, 26, 24, 12, 4, 1, 158, 130, 60, 20, 5, 1, 1330, 948, 390, 120, 30, 6, 1, 15414, 9310, 3318, 910, 210, 42, 7, 1, 245578, 123312, 37240, 8848, 1820, 336, 56, 8, 1, 5382862, 2210202, 554904, 111720, 19908, 3276, 504, 72, 9, 1

Offset: 0

Vladimir Kruchinin and Peter Luschny, Mar 08 2020

T(n,k) is the number of n X n matrices over F_2 in reduced row echelon form having exactly k zero-columns. Equivalently, T(n,k) is the number of subspaces of F_2^n that "involve" n-k coordinates. (For the definition of "involve" see the link below: D. E. Knuth, Letter to Daniel Ullman and others). - Geoffrey Critzer, May 03 2025

			Triangle starts:
[0] [1]
[1] [1,       1]
[2] [2,       2,       1]
[3] [6,       6,       3,      1]
[4] [26,      24,      12,     4,      1]
[5] [158,     130,     60,     20,     5,     1]
[6] [1330,    948,     390,    120,    30,    6,    1]
[7] [15414,   9310,    3318,   910,    210,   42,   7,   1]
[8] [245578,  123312,  37240,  8848,   1820,  336,  56,  8,  1]
[9] [5382862, 2210202, 554904, 111720, 19908, 3276, 504, 72, 9, 1]

D. E. Knuth, Letter to Daniel Ullman and others, Apr 29 1997 [Annotated scanned copy, with permission]
R. P. Stanley and S. C. Locke, Graphs without increasing paths: Solution to Problem 10572, The American Mathematical Monthly, 106(2) (1999), 168.

Row sums: A006116, first column: A135922.

Cf. A022166, A323842, A323843.

T := (n, k) -> local j, m; pochhammer(n - k + 1, k)*add((-1)^j*add(product((2^(i + m) - 1)/(2^i - 1), i = 1..n-k-m-j), m = 0..n-k-j)*binomial(n - k, j), j = 0..n-k) / k!: for n from 0 to 9 do seq(T(n,k), k=0..n) od;  # Peter Luschny, Oct 08 2023

T[n_,k_]:= (Pochhammer[n-k+1,k]/(k!)*Sum[(-1)^j*Sum[Product[(2^(i+m)-1)/(2^i-1),{i,1,n-k-m-j}],{m,0,n-k-j}]*Binomial[n-k,j],{j,0,n-k}]); Flatten[Table[T[n,k],{n,0,9},{k,0,n}]] (* Detlef Meya, Oct 07 2023 *)

R = PolynomialRing(QQ, 'x')
x = R.gen()
@cached_function
def P(n, k, x):
    if k < 0 or n < 0: return R(0)
    if k == 0: return R(1)
    return x*P(n, k-1, x) + 2^k*P(n-1, k, (x+1)/2)
def row(n): return sum(P(n-k, k, x) for k in range(n+1)).coefficients()
print(flatten([row(n) for n in range(10)]))

Let P(n, k, x) = x*P(n, k-1, x) + 2^k*P(n-1, k, (x+1)/2) and Q(n, x) = Sum_{k=0..n} P(n-k, k, x) then T(n, k) = [x^k] Q(n, x).
T(n, k) = (1/k!) * Pochhammer(n-k+1, k) * Sum_{j=0..n-k}((-1)^j*Sum_{m=0..n-k-j} (Product_{i=1..n-k-m-j} ((2^(i+m)-1)/(2^i-1))) * binomial(n-k, j)). - Detlef Meya, Oct 07 2023
T(n,k) = binomial(n,k)*A135922(n-k). (see Stanley-Locke link above) - Geoffrey Critzer, May 03 2025
E.g.f.: exp(y x)*f(x) where f(x) is the e.g.f. for A135922. - Geoffrey Critzer, May 03 2025

A383655 Triangle read by rows: T(n,k) is the number of n-node Stanley graphs containing exactly k isolated points, n>=0, 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 11, 8, 6, 0, 1, 72, 55, 20, 10, 0, 1, 677, 432, 165, 40, 15, 0, 1, 8686, 4739, 1512, 385, 70, 21, 0, 1, 152191, 69488, 18956, 4032, 770, 112, 28, 0, 1, 3632916, 1369719, 312696, 56868, 9072, 1386, 168, 36, 0, 1, 118317913, 36329160, 6848595, 1042320, 142170, 18144, 2310, 240, 45, 0, 1

Offset: 0

Geoffrey Critzer, May 04 2025

For precise definition see the links: David Bevan and others (2023) or D.E. Knuth (1997).

			Triangle T(n,k) begins:
   1;
   0,  1;
   1,  0,  1;
   2,  3,  0,  1;
  11,  8,  6,  0, 1;
  72, 55, 20, 10, 0, 1;
  ...

David Bevan, Gi-Sang Cheon and Sergey Kitaev, On naturally labelled posets and permutations avoiding 12-34, arXiv:2311.08023 [math.CO], 2023.
D. E. Knuth, Letter to Daniel Ullman and others, Apr 29 1997 [Annotated scanned copy, with permission].

Cf. A323842 (column k=0), A135922 (row sums).

nn = 10; g[x_] :=Total[Table[Sum[QBinomial[n, k, 2] x^n/n!, {k, 0, n}], {n, 0, nn}]]; Table[(Range[0, nn]! CoefficientList[Series[Exp[y x] Exp[-x] g[x] Exp[-x], {x, 0, nn}], {x, y}])[[i, 1 ;; i]], {i, 1, nn + 1}] // Grid

E.g.f.: exp((y-1)*x)*f(x) where f(x) is the e.g.f. for A135922.

A383656 Triangular array read by rows: T(n,k) is the number of n-node Stanley graphs containing exactly k connected components, n>=0, 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 3, 1, 0, 8, 11, 6, 1, 0, 52, 60, 35, 10, 1, 0, 502, 472, 255, 85, 15, 1, 0, 6824, 5166, 2422, 805, 175, 21, 1, 0, 127166, 76712, 30072, 9177, 2100, 322, 28, 1, 0, 3205924, 1526910, 486800, 129360, 28497, 4788, 546, 36, 1, 0, 108975934, 40603534, 10292970, 2285240, 455805, 76629, 9870, 870, 45, 1

Offset: 0

Geoffrey Critzer, May 04 2025

For precise definition see the links: David Bevan and others (2023) or D.E. Knuth (1997).

			Triangle begins:
 1;
 0, 1;
 0, 1, 1;
 0, 2, 3, 1;
 0, 8, 11, 6, 1;
 0, 52, 60, 35, 10, 1;
 0, 502, 472, 255, 85, 15, 1;
 ...

David Bevan, Gi-Sang Cheon, and Sergey Kitaev, On naturally labelled posets and permutations avoiding 12-34, arXiv:2311.08023 [math.CO], 2023.
D. E. Knuth, Letter to Daniel Ullman and others, Apr 29 1997 [Annotated scanned copy, with permission].

Cf. A323843 (column k=1), A135922 (row sums).

nn = 8; Prepend[Table[(Range[0, nn]! CoefficientList[Series[(Exp[-x] g[x])^y, {x, 0, nn}], {x, y}])[[i,1 ;; i]], {i, 2, nn + 1}], {1}] // Grid

E.g.f.: f(x)^y where f(x) is the e.g.f. for A135922.

cp's OEIS Frontend

A007889 Number of intransitive (or alternating, or Stanley) trees: vertices are [0,n] and for no i

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A135922 Inverse binomial transform of A006116, which is the sum of Gaussian binomial coefficients [n,k] for q=2.

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A323842 Number of n-node Stanley graphs without isolated nodes.

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Maxima

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A323843 Number of n-node connected Stanley graphs.

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A029847 Gessel-Stanley triangle read by rows: triangle of coefficients of polynomials arising in connection with enumeration of intransitive trees by number of nodes and number of right nodes.

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A329154 Coefficients of polynomials related to the sum of Gaussian binomial coefficients for q = 2. Triangle read by rows, T(n,k) for 0 <= k <= n.

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