A084262 -id:A084262 - cp's OEIS frontend

A132382 Lower triangular array T(n,k) generator for group of arrays related to A001147 and A102625.

1, -1, 1, -1, -2, 1, -3, -3, -3, 1, -15, -12, -6, -4, 1, -105, -75, -30, -10, -5, 1, -945, -630, -225, -60, -15, -6, 1, -10395, -6615, -2205, -525, -105, -21, -7, 1, -135135, -83160, -26460, -5880, -1050, -168, -28, -8, 1, -2027025, -1216215, -374220, -79380, -13230, -1890, -252, -36, -9, 1

Offset: 0

Tom Copeland, Nov 11 2007, Nov 12 2007, Nov 19 2007, Dec 04 2007, Dec 06 2007

Let b(n) = LPT[ A001147 ] = -A001147(n-1) for n > 0 and 1 for n=0, where LPT represents the action of the list partition transform described in A133314.
Then T(n,k) = binomial(n,k) * b(n-k) .
Form the matrix of polynomials TB(n,k,t) = T(n,k) * t^(n-k) = binomial(n,k) * b(n-k) * t^(n-k) = binomial(n,k) * Pb(n-k,t),
beginning as
     1;
    -1,       1;
    -1*t,    -2,       1;
    -3*t^2,  -3*t,    -3,       1;
   -15*t^3, -12*t^2,  -6*t,    -4,    1;
  -105*t^4, -75*t^3, -30*t^2, -10*t, -5, 1;
Let Pc(n,t) = LPT(Pb(.,t)).
Then [TB(t)]^(-1) = TC(t) = [ binomial(n,k) * Pc(n-k,t) ] = LPT(TB),
whose first column is
Pc(0,t) = 1
Pc(1,t) = 1
Pc(2,t) = 2 + t
Pc(3,t) = 6 + 6*t + 3*t^2
Pc(4,t) = 24 + 36*t + 30*t^2 + 15*t^3
Pc(5,t) = 120 + 240*t + 270*t^2 + 210*t^3 + 105*t^4.
The coefficients of these polynomials are given by the reverse of A102625 with the highest order coefficients given by A001147 with an additional leading 1.
Note this is not the complete matrix TC. The complete matrix is formed by multiplying along the diagonal of the lower triangular Pascal matrix by these polynomials, embedding trees of coefficients in the matrix.
exp[Pb(.,t)*x] = 1 + [(1-2t*x)^(1/2) - 1] / (t-0) = [1 + a finite diff. of [(1-2t*x)^(1/2)] with step t] = e.g.f. of the first column of TB.
exp[Pc(.,t)*x] = 1 / { 1 + [(1-2t*x)^(1/2) - 1] / t } = 1 / exp[Pb(.,t)*x) = e.g.f. of the first column of TC.
TB(t) and TC(t), being inverse to each other, are the generators of an Abelian group.
TB(0) and TC(0) are generators for a subgroup representing the iterated Laguerre operator described in A132013 and A132014.
Let sb(t,m) and sc(t,m) be the associated sequences under the LPT to TB(t)^m = B(t,m) and TC(t)^m = C(t,m).
Let Esb(t,m) and Esc(t,m) be e.g.f.'s for sb(t,m) and sc(t,m), rB(t,m) and rC(t,m) be the row sums of B(t,m) and C(t,m) and aB(t,m) and aC(t,m) be the alternating row sums.
Then B(t,m) is the inverse of C(t,m), Esb(t,m) is the reciprocal of Esc(t,m) and sb(t,m) and sc(t,m) form a reciprocal pair under the LPT. Similar relations hold among the row sums and the alternating sign row sums and associated quantities.
All the group members have the form B(t,m) * C(u,p) = TB(t)^m * TC(u)^p = [ binomial(n,k) * s(n-k) ]
with associated e.g.f. Es(x) = exp[m * Pb(.,t) * x] * exp[p * Pc(.,u) * x] for the first column of the matrix, with terms s(n), so group multiplication is isomorphic to matrix multiplication and to multiplication of the e.g.f.'s for the associated sequences (see examples).
These results can be extended to other groups of integer-valued arrays by replacing the 2 by any natural number in the expression for exp[Pb(.,t)*x].
More generally,
[ G.f. for M = Product_{i=0..j} B[s(i),m(i)] * C[t(i),n(i)] ]
= exp(u*x) * Product_{i=0..j} { exp[m(i) * Pb(.,s(i)) * x] * exp[n(i) * Pc(.,t(i)) * x] }
= exp(u*x) * Product_{i=0..j} { 1 + [ (1 - 2*s(i)*x)^(1/2) - 1 ] / s(i) }^m(i) / { 1 + [ (1 - 2*t(i)*x)^(1/2) - 1 ] / t(i) }^n(i)
= exp(u*x) * H(x)
[ E.g.f. for M ] = I_o[2*(u*x)^(1/2)] * H(x).
M is an integer-valued matrix for m(i) and n(i) positive integers and s(i) and t(i) integers. To invert M, change B to C in Product for M.
H(x) is the e.g.f. for the first column of M and diagonally multiplying the Pascal matrix by the terms of this column generates M. See examples.
The G.f. for M, i.e., the e.g.f. for the row polynomials of M, implies that the row polynomials form an Appell sequence (see Wikipedia and Mathworld). - Tom Copeland, Dec 03 2013

			Some group members and associated arrays are
(t,m) :: Array :: Asc. Matrix :: Asc. Sequence :: E.g.f. for sequence
..............................................................................
(0,1).::.B..::..A132013.::.(1,-1,0,0,0,0,...).....::.s(x).=.1-x
(0,1).::.C..::..A094587.::.(0!,1!,2!,3!,...)......::.1./.s(x)
(0,1).::.rB.::.~A055137.::.(1,0,-1,-2,-3,-4,...)..::.exp(x).*.s(x)
(0,1).::.rC.::....-.....::..A000522...............::.exp(x)./.s(x)
(0,1).::.aB.::....-.....::.(1,-2,3,-4,5,-6,...)...::.exp(-x).*.s(x)
(0,1).::.aC.::..A008290.::..A000166...............::.exp(-x)./.s(x)
..............................................................................
(0,2).::.B..::..A132014.::.(1,-2,2,0,0,0,0...)....::.s(x).=.(1-x)^2
(0,2).::.C..::..A132159.::.(1!,2!,3!,4!,...)......::..1./.s(x).
(0,2).::.rB.::...-......::.(1,-1,-1,1,5,11,19,29,)::.exp(x).*.s(x).
(0,2).::.rC.::...-......::..A001339...............::.exp(x)./.s(x).
(0,2).::.aB.::...-......::.(-1)^n.A002061(n+1)....::.exp(-x).*.s(x).
(0,2).::.aC.::...-......::..A000255...............::.exp(-x)./.s(x).
..............................................................................
(1,1).::.B..::..T.......::.(1,-A001147(n-1))......::.s(x).=.(1-2x)^(1/2)
(1,1).::.C..::.~A113278.::..A001147...............::.1./.s(x)...
(1,1).::.rB.::...-......::..A055142...............::.exp(x).*.s(x).
(1,1).::.rC.::...-......::..A084262...............::.exp(x)./.s(x).
(1,1).::.aB.::...-......::.(1,-2,2,-4,-4,-56,...).::.exp(-x).*.s(x).
(1,1).::.aC.::...-......::..A053871...............::.exp(-x)./.s(x).
..............................................................................
(2,1).::.B..::...-......::.(1,-A001813)...........::.s=[1+(1-4x)^(1/2)]/2....
(2,1).::.C..::...-......::..A001761...............::.1./.s(x)..
(2,1).::.rB.::...-......::.(1,0,-3,-20,-183,...)..::.exp(x).*.s(x)..
(2,1).::.rC.::...-......::.(1,2,7,46,485,...).....::.exp(x)./.s(x).
(2,1).::.aB.::...-......::.(1,-2,1,-10,-79,...)...::.exp(-x).*.s(x).
(2,1).::.aC.::...-......::.(1,0,3,20,237,...).....::.exp(-x)./.s(x)
..............................................................................
(1,2).::.B..::.~A134082.::.(1,-2,0,0,0,0,...).....::.s(x).=.1.-.2x
(1,2).::.C..::....-.....::..A000165...............::.1./.s(x)..
(1,2).::.rB.::....-.....::.(1,-1,-3,-5,-7,-9,...).::.exp(x).*.s(x).
(1,2).::.rC.::....-.....::..A010844...............::.exp(x)./.s(x)..
(1,2).::.aB.::....-.....::.(1,-3,5,-7,9,-11,...)..::.exp(-x).*.s(x).
(1,2).::.aC.::....-.....::..A000354...............::.exp(-x)./.s(x).
..............................................................................
(The tilde indicates the match is not exact--specifically, there are differences in signs from the true matrices.)
Note the row sums correspond to binomial transforms of s(x) and the alternating row sums, to inverse binomial transforms, or, finite differences.
Some additional examples:
C(1,2)*B(0,1) = B(1,-2)*C(0,-1) = [ binomial(n,k)*A002866(n-k) ] with asc. e.g.f. (1-x) / (1-2x).
B(1,2)*C(0,1) = C(1,-2)*B(0,-1) = 2I - A094587 with asc. e.g.f. (1-2x) / (1-x).

[G.f. for TB(n,k,t)] = GTB(u,x,t) = exp(u*x) * { 1 + [ (1 - 2t*x)^(1/2) - 1 ] / t } = exp[(u+Pb(.,t))*x] where TB(n,k,t) = (D_x)^n (D_u)^k /k! GTB(u,x,t) eval. at u=x=0.
[G.f. for TC(n,k,t)] = GTC(u,x,t) = exp(u*x) / { 1 + [ (1 - 2t*x)^(1/2) - 1 ] / t } = exp[(u+Pc(.,t))*x] where TC(n,k,t) = (D_x)^n (D_u)^k /k! GTC(u,x,t) eval. at u=x=0.
[E.g.f. for TB(n,k,t)] = I_o[2*(u*x)^(1/2)] * { 1 + [ (1 - 2t*x)^(1/2) - 1 ] / t } and
[E.g.f. for TC(n,k,t)] = I_o[2*(u*x)^(1/2)] / { 1 + [ (1 - 2t*x)^(1/2) - 1 ] / t }
where I_o is the zeroth modified Bessel function of the first kind, i.e.,
I_o[2*(u*x)^(1/2)] = Sum_{j>=0} (u^j/j!) * (x^j/j!).
So [e.g.f. for TB(n,k)] = I_o[2*(u*x)^(1/2)] * (1 - 2x)^(1/2).

More terms from Tom Copeland, Dec 05 2007

A295099 a(n) = n! * [x^n] exp(nx)/sqrt(1 - 2x).

Original entry on oeis.org

1, 2, 11, 96, 1145, 17320, 317547, 6843872, 169603793, 4752704160, 148631984075, 5132717953792, 194022218612169, 7969667589513344, 353510496652374635, 16842274069331520000, 857827370723082312737, 46516913938434654949888, 2675772791433589181094027, 162742831545094476694814720

Offset: 0

Ilya Gutkovskiy, Nov 14 2017

nonn

The n-th term of the n-th binomial transform of A001147.

G. C. Greubel, Table of n, a(n) for n = 0..250
N. J. A. Sloane, Transforms
Index entries for sequences related to factorial numbers

Cf. A001147, A063170, A081367, A084262, A295098, A295100.

Table[n! SeriesCoefficient[Exp[n x]/Sqrt[1 - 2 x], {x, 0, n}], {n, 0, 19}]

a(n) ~ 2^(n+1) * n^n / exp(n/2). - Vaclav Kotesovec, Nov 14 2017

A346258 E.g.f.: exp(x) / (1 - 3 * x)^(1/3).

Original entry on oeis.org

1, 2, 7, 44, 421, 5366, 84907, 1601552, 35052649, 872931626, 24368595631, 753607111412, 25572085243597, 944609383245854, 37731673388579731, 1620520035001182296, 74466516342569480017, 3645540855448417250642, 189415873005295070803159, 10410429682102309433442236

Offset: 0

Ilya Gutkovskiy, Aug 10 2021

Binomial transform of A007559.

Cf. A000522, A007559, A010845, A033030, A084262, A347012, A347013, A347014.

g:= proc(n) option remember; `if`(n<2, 1, (3*n-2)*g(n-1)) end:
a:= n-> add(binomial(n, k)*g(k), k=0..n):
seq(a(n), n=0..19);  # Alois P. Heinz, Aug 10 2021

nmax = 19; CoefficientList[Series[Exp[x]/(1 - 3 x)^(1/3), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k] 3^k Pochhammer[1/3, k], {k, 0, n}], {n, 0, 19}]
Table[HypergeometricU[1/3, n + 4/3, 1/3]/3^(1/3), {n, 0, 19}]
With[{nn=20},CoefficientList[Series[Exp[x]/CubeRoot[1-3x],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 25 2025 *)

a[n]:=if n<2 then n+1 else (3*n-1)*a[n-1]+3*(1-n)*a[n-2];
makelist(a[n],n,0,50); /* Tani Akinari, Sep 08 2023 */

a(n) = Sum_{k=0..n} binomial(n,k) * A007559(k).
a(n) ~ n! * exp(1/3) * 3^n / (Gamma(1/3) * n^(2/3)). - Vaclav Kotesovec, Aug 14 2021
a(n+2) = (3*n+5)*a(n+1)-3*(n+1)*a(n). - Tani Akinari, Sep 08 2023

A347012 E.g.f.: exp(x) / (1 - 4 * x)^(1/4).

Original entry on oeis.org

1, 2, 8, 64, 800, 13376, 278272, 6914048, 199629824, 6566164480, 242327576576, 9915111636992, 445432721932288, 21795710738038784, 1153805878313615360, 65700181140859518976, 4004182878034473254912, 260071258357260225609728, 17932703649301871611346944

Offset: 0

Ilya Gutkovskiy, Aug 10 2021

nonn

Binomial transform of A007696.

Cf. A000522, A007696, A056545, A084262, A088991, A346258, A347013, A347014.

g:= proc(n) option remember; `if`(n<2, 1, (4*n-3)*g(n-1)) end:
a:= n-> add(binomial(n, k)*g(k), k=0..n):
seq(a(n), n=0..18);  # Alois P. Heinz, Aug 10 2021

nmax = 18; CoefficientList[Series[Exp[x]/(1 - 4 x)^(1/4), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k] 4^k Pochhammer[1/4, k], {k, 0, n}], {n, 0, 18}]
Table[HypergeometricU[1/4, n + 5/4, 1/4]/Sqrt[2], {n, 0, 18}]

a(n) = Sum_{k=0..n} binomial(n,k) * A007696(k).

a(n) ~ n! * exp(1/4) * 4^n / (Gamma(1/4) * n^(3/4)). - Vaclav Kotesovec, Aug 14 2021

A347013 E.g.f.: exp(x) / (1 - 5 * x)^(1/5).

Original entry on oeis.org

1, 2, 9, 88, 1361, 28182, 726889, 22414988, 803913441, 32867765002, 1508608850249, 76804271962848, 4294870015118641, 261673684619584862, 17252970318529474089, 1223896705010751194068, 92946073511938131386561, 7523666291578066678172562, 646658551118777059833155209

Offset: 0

Ilya Gutkovskiy, Aug 10 2021

nonn

Binomial transform of A008548.

Cf. A000522, A008548, A056546, A084262, A088992, A346258, A347012, A347014.

g:= proc(n) option remember; `if`(n<2, 1, (5*n-4)*g(n-1)) end:
a:= n-> add(binomial(n, k)*g(k), k=0..n):
seq(a(n), n=0..18);  # Alois P. Heinz, Aug 10 2021

nmax = 18; CoefficientList[Series[Exp[x]/(1 - 5 x)^(1/5), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k] 5^k Pochhammer[1/5, k], {k, 0, n}], {n, 0, 18}]
Table[HypergeometricU[1/5, n + 6/5, 1/5]/5^(1/5), {n, 0, 18}]

a(n) = Sum_{k=0..n} binomial(n,k) * A008548(k).

a(n) ~ n! * exp(1/5) * 5^n / (Gamma(1/5) * n^(4/5)). - Vaclav Kotesovec, Aug 14 2021

A347014 Expansion of e.g.f.: exp(x) / (1 - 6*x)^(1/6).

Original entry on oeis.org

1, 2, 10, 116, 2140, 52856, 1627192, 59930480, 2568599056, 125553289760, 6892279877536, 419788155021632, 28090704069553600, 2048487353985408896, 161687913401407530880, 13733087614786273308416, 1248892148354210466595072, 121073054127693143488709120

Offset: 0

Ilya Gutkovskiy, Aug 10 2021

nonn

Binomial transform of A008542.

In general, for k >= 1, if e.g.f. = exp(x) / (1 - k*x)^(1/k), then a(n) ~ n! * exp(1/k) * k^n / (Gamma(1/k) * n^(1 - 1/k)). - Vaclav Kotesovec, Aug 14 2021

Cf. A000522, A008542, A056547, A084262, A346258, A347012, A347013.

g:= proc(n) option remember; `if`(n<2, 1, (6*n-5)*g(n-1)) end:
a:= n-> add(binomial(n, k)*g(k), k=0..n):
seq(a(n), n=0..17);  # Alois P. Heinz, Aug 10 2021

nmax = 17; CoefficientList[Series[Exp[x]/(1 - 6 x)^(1/6), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k] 6^k Pochhammer[1/6, k], {k, 0, n}], {n, 0, 17}]
Table[HypergeometricU[1/6, n + 7/6, 1/6]/6^(1/6), {n, 0, 17}]

a[n]:=if n<2 then n+1 else (6*n-4)*a[n-1]-6*(n-1)*a[n-2];
makelist(a[n],n,0,50); /* Tani Akinari, Sep 08 2023 */

a(n) = Sum_{k=0..n} binomial(n,k) * A008542(k).
a(n) ~ n! * exp(1/6) * 6^n / (Gamma(1/6) * n^(5/6)). - Vaclav Kotesovec, Aug 14 2021
a(n+2) = (6*n+8)*a(n+1) - 6*(n+1)*a(n). - Tani Akinari, Sep 08 2023

A220452 Number of unordered full binary trees with labels from a set of n labels.

Original entry on oeis.org

1, 3, 9, 37, 225, 1881, 19873, 251889, 3712257, 62286625, 1171487361, 24402416193, 557542291969, 13861636770177, 372514645389825, 10759590258589441, 332386419622387713, 10935312198369141249, 381705328034883127297, 14089260601787531469825, 548302210950105933701121

Offset: 1

Felix A. Pahl, Dec 15 2012

a(n) is the size of the population generated by n unrelated ancestors if two individuals produce one descendant together if and only if they are not related.

			For n=3, each of the three pairs of ancestors produces one descendant, and each of these descendants produces one more descendant with the respective remaining ancestor; three ancestors, three first-order descendants and three second-order descendants makes a population of a(3)=9.

Felix A. Pahl, Table of n, a(n) for n = 1..400
Mathematics Stack Exchange, The n immortals problem
V. Kotesovec, Interesting asymptotic formulas for binomial sums, Jun 09 2013

Cf. A001147.

Partial sums of A084262.

import java.math.BigInteger;
public class A220452 {
    public static void main (String [] args) {
        int max = Integer.parseInt (args [0]);
        BigInteger [] doubleFactorials = new BigInteger [max + 1];
        BigInteger [] [] binomialCoefficients = new BigInteger [max + 1] [max + 1];
        doubleFactorials [0] = BigInteger.ONE;
        for (int n = 1;n <= max;n++) {
            binomialCoefficients [n] [0] = BigInteger.ONE;
            BigInteger sum = BigInteger.ZERO;
            for (int k = 1;k <= n;k++) {
                binomialCoefficients [n] [k] = k == n ? BigInteger.ONE : binomialCoefficients [n - 1] [k - 1].add (binomialCoefficients [n - 1] [k]);
                sum = sum.add (binomialCoefficients [n] [k].multiply (doubleFactorials [k - 1]));
            }
            System.out.println (n + " " + sum);
            doubleFactorials [n] = doubleFactorials [n - 1].multiply (BigInteger.valueOf (2 * n - 1));
        }
    }
}

Table[Sum[Binomial[n,k]*(2*k-3)!!, {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Dec 17 2012 *)

a(n) = Sum_{k=1..n} binomial(n,k)*(2k-3)!!.

a(n) ~ (2n-3)!!*sqrt(e) ~ (2n)!/(n!*2^n*(2n-1))*sqrt(e) ~ n^(n-1)*2^(n-1/2)*exp(1/2-n). - Vaclav Kotesovec, Dec 17 2012

A293469 a(n) = Sum_{k=0..n} (2k-1)!!binomial(2*n-k, n).

Original entry on oeis.org

1, 3, 12, 57, 330, 2436, 23226, 277389, 3966534, 65517210, 1220999208, 25279328958, 575024187192, 14247595540542, 381846383109030, 11004598454925405, 339324532631899110, 11146022446431209490, 388535338484934710040, 14324570939127320452350, 556887682690152668745660

Offset: 0

Ilya Gutkovskiy, Oct 09 2017

nonn

Robert Israel, Table of n, a(n) for n = 0..403
Index entries for sequences related to factorial numbers

Cf. A001147, A076795, A084262, A092392, A270447, A293468.

seq(add(doublefactorial(2*k-1)*binomial(2*n-k,n),k=0..n),n=0..40); # Robert Israel, Oct 09 2017

Table[Sum[(2 k - 1)!! Binomial[2 n - k, n], {k, 0, n}], {n, 0, 20}]
Table[SeriesCoefficient[(1/(1 - x)^(n + 1)) 1/(1 + ContinuedFractionK[-k x, 1, {k, 1, n}]), {x, 0, n}], {n, 0, 20}]
Table[SeriesCoefficient[(1/(1 - x)^(n + 1)) Sum[(2 k - 1)!! x^k, {k, 0, n}], {x, 0, n}], {n, 0, 20}]

a(n) = [x^n] 1/((1 - x)^(n+1)*(1 - x/(1 - 2*x/(1 - 3*x/(1 - 4*x/(1 - 5*x/(1 - 6*x/(1 - ...)))))))), a continued fraction.
a(n) = Gamma(n+1/2)*hypergeom([1/2, 1, -n], [-2*n], 2)*4^n/(n!*sqrt(Pi)). - Robert Israel, Oct 09 2017
a(n) ~ 2^(n + 1/2) * n^n / exp(n - 1/2). - Vaclav Kotesovec, Oct 18 2017

A317618 Expansion of e.g.f. sqrt((1 - x)/(1 - 3*x)).

Original entry on oeis.org

1, 1, 5, 39, 417, 5685, 94365, 1847475, 41686785, 1065288105, 30411314325, 959236098975, 33129890726625, 1243507150410525, 50401090111697325, 2193907232242600875, 102075654396429338625, 5055304328553234380625, 265522264682686831945125, 14742355948224269570580375

Offset: 0

Ilya Gutkovskiy, Aug 01 2018

nonn

Lah transform of A001147.

Robert Israel, Table of n, a(n) for n = 0..380
N. J. A. Sloane, Transforms

Cf. A000246, A001147, A002866, A052563, A084262.

a:=series(sqrt((1 - x)/(1 - 3*x)), x=0, 20): seq(n!*coeff(a, x, n), n=0..19); # Paolo P. Lava, Mar 26 2019

nmax = 19; CoefficientList[Series[Sqrt[(1 - x)/(1 - 3*x)], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n - 1, k - 1] (2 k - 1)!! n!/k!, {k, 0, n}], {n, 0, 19}]
Join[{1}, Table[n! Hypergeometric2F1[3/2, 1 - n, 2, -2], {n, 19}]]

my(x='x + O('x^25)); Vec(serlaplace(sqrt((1 - x)/(1 - 3*x)))) \\ Michel Marcus, Mar 26 2019

a(n) = Sum_{k=0..n} binomial(n-1,k-1)*(2*k-1)!!*n!/k!.
a(n) ~ 2 * 3^(n - 1/2) * n^n / exp(n). - Vaclav Kotesovec, Mar 26 2019
D-finite with recurrence: (3*n^2 + 3*n)*a(n) + (-5 - 4*n)*a(n + 1) + a(n + 2)=0. - Robert Israel, Mar 26 2019

A343804 T(n, k) = Sum_{j=k..n} binomial(n, j)*E2(j, j-k), where E2 are the Eulerian numbers A201637. Triangle read by rows, T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 15, 11, 1, 1, 64, 96, 26, 1, 1, 325, 824, 448, 57, 1, 1, 1956, 7417, 6718, 1779, 120, 1, 1, 13699, 71595, 96633, 43411, 6429, 247, 1, 1, 109600, 746232, 1393588, 944618, 243928, 21898, 502, 1, 1, 986409, 8403000, 20600856, 19521210, 7739362, 1250774, 71742, 1013, 1

Offset: 0

Peter Luschny, Apr 30 2021

			Triangle starts:
[0] 1
[1] 1, 1
[2] 1, 4,      1
[3] 1, 15,     11,      1
[4] 1, 64,     96,      26,       1
[5] 1, 325,    824,     448,      57,       1
[6] 1, 1956,   7417,    6718,     1779,     120,     1
[7] 1, 13699,  71595,   96633,    43411,    6429,    247,     1
[8] 1, 109600, 746232,  1393588,  944618,   243928,  21898,   502,   1
[9] 1, 986409, 8403000, 20600856, 19521210, 7739362, 1250774, 71742, 1013, 1

Row sums: A084262.

Cf. A046802 (Eulerian first order).

T := (n, k) -> add(binomial(n, r)*combinat:-eulerian2(r, r-k), r = k..n):
seq(seq(T(n, k), k = 0..n), n = 0..9);

cp's OEIS Frontend

A132382 Lower triangular array T(n,k) generator for group of arrays related to A001147 and A102625.

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A295099 a(n) = n! * [x^n] exp(n*x)/sqrt(1 - 2*x).

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A346258 E.g.f.: exp(x) / (1 - 3 * x)^(1/3).

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Maple

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Maxima

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A347012 E.g.f.: exp(x) / (1 - 4 * x)^(1/4).

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Maple

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A347013 E.g.f.: exp(x) / (1 - 5 * x)^(1/5).

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A347014 Expansion of e.g.f.: exp(x) / (1 - 6*x)^(1/6).

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Maple

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Maxima

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A220452 Number of unordered full binary trees with labels from a set of n labels.

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A293469 a(n) = Sum_{k=0..n} (2*k-1)!!*binomial(2*n-k, n).

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A295099 a(n) = n! * [x^n] exp(nx)/sqrt(1 - 2x).

A293469 a(n) = Sum_{k=0..n} (2k-1)!!binomial(2*n-k, n).