A055372 -id:A055372 - cp's OEIS frontend

A081294 Expansion of (1-2x)/(1-4x).

1, 2, 8, 32, 128, 512, 2048, 8192, 32768, 131072, 524288, 2097152, 8388608, 33554432, 134217728, 536870912, 2147483648, 8589934592, 34359738368, 137438953472, 549755813888, 2199023255552, 8796093022208, 35184372088832

Offset: 0

Paul Barry, Mar 17 2003

Binomial transform of A046717. Second binomial transform of A000302 (with interpolated zeros). Partial sums are A007583.
Counts closed walks of length 2n at a vertex of the cyclic graph on 4 nodes C_4. With interpolated zeros, counts closed walks of length n at a vertex of the cyclic graph on 4 nodes C_4. - Paul Barry, Mar 10 2004
In general, Sum_{k=0..n} Sum_{j=0..n} C(2*(n-k), j)*C(2*k, j)*r^j has expansion (1 - (r+1)*x)/(1 - (r+3)*x - (r-1)*(r+3)*x^2 + (r-1)^3*x^3). - Paul Barry, Jun 04 2005 [corrected by Jason Yuen, Jan 20 2025]
a(n) is the number of binary strings of length 2n with an even number of 0's (and hence an even number of 1's). - Toby Gottfried, Mar 22 2010
Number of compositions of n where there are 2 sorts of part 1, 4 sorts of part 2, 8 sorts of part 3, ..., 2^k sorts of part k. - Joerg Arndt, Aug 04 2014
a(n) is also the number of permutations simultaneously avoiding 231 and 321 in the classical sense which can be realized as labels on an increasing strict binary tree with 2n-1 nodes. See A245904 for more information on increasing strict binary trees. - Manda Riehl Aug 07 2014
INVERT transform of powers of 2 (A000079). - Alois P. Heinz, Feb 11 2021
a(n) is the number of elements in an n-interval of the binomial poset of even-sized subsets of positive integers, cf. Stanley reference and second formula by Paul Barry.  Each multichain 0 = x_0 <= x_1 <= x_2 = 1 in such an n-interval corresponds to a closed walk described above by Paul Barry.  More generally, each multichain 0 = x_0 <= x_1 <= ... <= x_k = 1 corresponds to a closed walk of length 2n on the k-dimensional hypercube, cf. A054879, A092812, A121822. - Geoffrey Critzer, Apr 21 2023

			G.f. = 1 + 2*x + 8*x^2 + 32*x^3 + 128*x^4 + 512*x^5 + 2048*x^6 + 8192*x^7 + ...

Richard P. Stanley, Enumerative Combinatorics, Vol 1, second edition, Example 3.18.3-f, page 323.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
M. Paukner, L. Pepin, M. Riehl, and J. Wieser, Pattern Avoidance in Task-Precedence Posets, arXiv:1511.00080 [math.CO], 2015-2016.
Index to divisibility sequences.
Index entries for linear recurrences with constant coefficients, signature (4).

Row sums of triangle A136158.

Cf. A000079, A081295, A009117, A016742, A054879, A092812, A121822. Essentially the same as A004171.

[(4^n+0^n)/2: n in [0..30]]; // Vincenzo Librandi, Jul 26 2011

R:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( (1-2*x)/(1-4*x))); // Marius A. Burtea, Jan 20 2020

a:= n-> 2^max(0, (2*n-1)):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 20 2017

CoefficientList[Series[(1-2x)/(1-4x),{x,0,40}],x] (* or *)
Join[{1}, NestList[4 # &, 2, 40]] (* Harvey P. Dale, Apr 22 2011 *)

a(n)=1<Charles R Greathouse IV, Jul 25 2011

x='x+O('x^100); Vec((1-2*x)/(1-4*x)) \\ Altug Alkan, Dec 21 2015

G.f.: (1-2*x)/(1-4*x).
a(n) = 4*a(n-1) n > 1, with a(0)=1, a(1)=2.
a(n) = (4^n+0^n)/2 (i.e., 1 followed by 4^n/2, n > 0).
E.g.f.: exp(2*x)*cosh(2*x) = (exp(4*x)+exp(0))/2. - Paul Barry, May 10 2003
a(n) = Sum_{k=0..n} C(2*n, 2*k). - Paul Barry, May 20 2003
a(n) = A001045(2*n+1) - A001045(2*n-1) + 0^n/2. - Paul Barry, Mar 10 2004
a(n) = 2^n*A011782(n); a(n) = gcd(A011782(2n), A011782(2n+1)). - Paul Barry, Jan 12 2005
a(n) = Sum_{k=0..n} Sum_{j=0..n} C(2*(n-k), j)*C(2*k, j). - Paul Barry, Jun 04 2005
a(n) = Sum_{k=0..n} A038763(n,k). - Philippe Deléham, Sep 22 2006
a(n) = Integral_{x=0..4} p(n,x)^2/(Pi*sqrt(x(4-x))) dx, where p(n,x) is the sequence of orthogonal polynomials defined by C(2*n,n): p(n,x) = (2*x-4)*p(n-1,x) - 4*p(n-2,x), with p(0,x)=1, p(1,x)=-2+x. - Paul Barry, Mar 01 2007
a(n) = ((2+sqrt(4))^n + (2-sqrt(4))^n)/2. - Al Hakanson (hawkuu(AT)gmail.com), Nov 22 2008
a(n) = A000079(n) * A011782(n). - Philippe Deléham, Dec 01 2008
a(n) = A004171(n-1) = A028403(n) - A000079(n) for n >= 1. - Jaroslav Krizek, Jul 27 2009
a(n) = Sum_{k=0..n} A201730(n,k)*3^k. - Philippe Deléham, Dec 06 2011
a(n) = Sum_{k=0..n} A134309(n,k)*2^k = Sum_{k=0..n} A055372(n,k). - Philippe Deléham, Feb 04 2012
G.f.: Q(0), where Q(k) = 1 - 2*x/(1 - 2/(2 - 1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Apr 29 2013
E.g.f.: 1/2 + exp(4*x)/2 = (Q(0)+1)/2, where Q(k) = 1 + 4*x/(2*k+1 - 2*x*(2*k+1)/(2*x + (k+1)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Apr 29 2013
a(n) = ceiling( 2^(2n-1) ). - Wesley Ivan Hurt, Jun 30 2013
G.f.: 1 + 2*x/(1 + x)*( 1 + 5*x/(1 + 4*x)*( 1 + 8*x/(1 + 7*x)*( 1 + 11*x/(1 + 10*x)*( 1 + ... )))). - Peter Bala, May 27 2017
Sum_{n>=0} 1/a(n) = 5/3. - Amiram Eldar, Aug 18 2022
Sum_{n>=0} a(n)*x^n/A000680(n) = E(x)^2 where E(x) = Sum_{n>=0} x^n/A000680(n). - Geoffrey Critzer, Apr 21 2023

A093143 Expansion of (1-5x)/(1-10x).

Original entry on oeis.org

1, 5, 50, 500, 5000, 50000, 500000, 5000000, 50000000, 500000000, 5000000000, 50000000000, 500000000000, 5000000000000, 50000000000000, 500000000000000, 5000000000000000, 50000000000000000, 500000000000000000, 5000000000000000000, 50000000000000000000, 500000000000000000000

Offset: 0

Paul Barry, Mar 24 2004

Partial sums are A093142. A convex combination of 10^n and 0^n.
a(n) is the number of compositions of even natural numbers in n parts <= 9 (0 is counted as a part); also the number of ways of placing of an even number of indistinguishable objects into n distinguishable boxes with the condition that at most 9 objects can be placed in each box. - Adi Dani, May 17 2011
See an A246057 comment with a reference for the k-family satisfying a so-called curious cubic identity involving A246057(k-1), a(k) and A002277(k). - Wolfdieter Lang, Feb 07 2017

			From _Adi Dani_, May 17 2011: (Start)
a(2)=50: there are 50 compositions of even numbers into 2 parts <= 9:
(0,0);
(0,2),(2,0),(1,1);
(0,4),(4,0),(1,3),(3,1),(2,2);
(0,6),(6,0),(1,5),(5,1),(2,4),(4,2),(3,3);
(0,8),(8,0),(1,7),(7,1),(2,6),(6,2),(3,5),(5,3),(4,4);
(1,9),(9,1),(2,8),(8,2),(3,7),(7,3),(4,6),(6,4),(5,5);
(3,9),(9,3),(4,8),(8,4),(5,7),(7,5),(6,6);
(5,9),(9,5),(6,8),(8,6),(7,7);
(7,9),(9,7),(8,8);
(9,9).
(End)
Curious cubic identities (see a comment above): 1^3 + 5^3 + 3^3 = 153, 16^3 + 50^3 + 33^3 = 165033, 166^3 + 500^3 + 333^3 = 166500333, ... - _Wolfdieter Lang_, Feb 07 2017

Adi Dani, Restricted compositions of natural numbers
Index entries for linear recurrences with constant coefficients, signature (10).

Cf. A002277, A055372, A093142, A134309, A246057.

Table[Ceiling[1/2*10^n],{n,0,30}] (* Adi Dani, Jun 20 2011 *)
Join[{1},NestList[10#&,5,20]] (* Harvey P. Dale, Apr 10 2021 *)

Vec((1-5*x)/(1-10*x) + O(x^100)) \\ Altug Alkan, Nov 01 2015

a(n) = 5*10^n/10 for n > 0.
a(n) = Sum_{k=0..n} A134309(n,k)*5^k = Sum_{k=0..n} A055372(n,k)*4^k. - Philippe Deléham, Feb 04 2012
From Elmo R. Oliveira, Aug 21 2024: (Start)
E.g.f.: (exp(10*x) + 1)/2.
a(n) = 10*a(n-1) for n > 1. (End)

a(19)-a(21) from Elmo R. Oliveira, Aug 21 2024

A081341 Expansion of exp(3x)cosh(3*x).

Original entry on oeis.org

1, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368, 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568

Offset: 0

Paul Barry, Mar 18 2003

Binomial transform of A081340. 3rd binomial transform of (1,0,9,0,81,0,729,0,...).
For m > 1, n > 0, A166469(A002110(m)*a(n)) = (n+1)*A000045(m+1). For n > 0, A166469(a(n)) = 2n. - Matthew Vandermast, Nov 05 2009
Number of compositions of even natural numbers in n parts <= 5. - Adi Dani, May 29 2011

			From _Adi Dani_, May 29 2011: (Start)
a(2)=18: there are 18 compositions of even natural numbers into 2 parts <= 5:
  for 0: (0,0);
  for 2: (0,2),(2,0),(1,1);
  for 4: (0,4),(4,0),(1,3),(3,1),(2,2);
  for 6: (1,5),(5,1),(2,4),(4,2),(3,3);
  for 8: (3,5),(5,3),(4,4);
  for 10: (5,5).  (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..125
Index entries for linear recurrences with constant coefficients, signature (6).

Cf. A000244, A034494, A081340, A081342.

a:= proc(n) option remember; `if`(n=0, 1,
      add(3^j*a(n-j), j=1..n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 22 2017

Table[Ceiling[1/2(6^n)], {n, 0, 25}]
CoefficientList[Series[-(-1 + 3 x)/(1 - 6 x), {x, 0, 50}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 21 2011 *)
Join[{1},NestList[6#&,3,30]] (* Harvey P. Dale, May 25 2019 *)

x='x+O('x^66); /* that many terms */
Vec((1-3*x)/(1-6*x)) /* show terms */ /* Joerg Arndt, May 29 2011 */

a(0)=1, a(n) = 6^n/2, n > 0.
G.f.: (1-3*x)/(1-6*x).
E.g.f.: exp(3*x)*cosh(3*x).
a(n) = A000244(n)*A011782(n). - Philippe Deléham, Dec 01 2008
a(n) = ((3+sqrt(9))^n + (3-sqrt(9))^n)/2. - Al Hakanson (hawkuu(AT)gmail.com), Dec 08 2008
a(n) = Sum_{k=0..n} A134309(n,k)*3^k = Sum_{k=0..n} A055372(n,k)*2^k. - Philippe Deléham, Feb 04 2012
From Sergei N. Gladkovskii, Jul 19 2012: (Start)
a(n) = ((8*n-4)*a(n-1) - 12*(n-2)*a(n-2))/n, a(0)=1, a(1)=3.
E.g.f. (exp(6*x) + 1)/2 = 1 + 3*x/(G(0) - 6*x) where G(k) = 6*x + 1 + k - 6*x*(k+1)/G(k+1) (continued fraction, Euler's 1st kind, 1-step). (End)
"INVERT" transform of A000244. - Alois P. Heinz, Sep 22 2017

Typo in A-number fixed by Klaus Brockhaus, Apr 04 2010

A134309 Triangle read by rows, where row n consists of n zeros followed by 2^(n-1).

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 32, 0, 0, 0, 0, 0, 0, 0, 64, 0, 0, 0, 0, 0, 0, 0, 0, 128, 0, 0, 0, 0, 0, 0, 0, 0, 0, 256, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 512, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1024, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2048, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Gary W. Adamson, Oct 19 2007

As infinite lower triangular matrices, binomial transform of A134309 = A082137. A134309 * A007318 = A055372. A134309 * [1,2,3,...] = A057711: (1, 2, 6, 16, 40, 96, 224,...).

Triangle read by rows given by [0,0,0,0,0,0,0,0,...] DELTA [1,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 20 2007

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  0, 1;
  0, 0, 2;
  0, 0, 0, 4;
  0, 0, 0, 0, 8;
  0, 0, 0, 0, 0, 16;
  ...

Cf. A011782 (diagonal elements: 1 followed by 1, 2, 4, 8, ... = A000079: 2^n).

Cf. A055372, A057711, A082137.

Join[{1},Flatten[Table[Join[{PadRight[{},n],2^(n-1)}],{n,20}]]] (* Harvey P. Dale, Jan 04 2024 *)

A134309(r,c)=if(r==c,2^max(r-1,0),0) \\ M. F. Hasler, Mar 29 2022

Triangle, T(0,0) = 1, then for n > 0, n zeros followed by 2^(n-1). Infinite lower triangular matrix with (1, 1, 2, 4, 8, 16, ...) in the main diagonal and the rest zeros.
G.f.: (1 - y*x)/(1 - 2*y*x). - Philippe Deléham, Feb 04 2012
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A011782(n), A081294(n), A081341(n), A092811(n), A093143(n), A067419(n) for x = 0, 1, 2, 3, 4, 5, 6 respectively. - Philippe Deléham, Feb 04 2012
Diagonal is A011782, other elements are 0. - M. F. Hasler, Mar 29 2022

A067419 Fourth column of triangle A067417.

Original entry on oeis.org

1, 6, 72, 864, 10368, 124416, 1492992, 17915904, 214990848, 2579890176, 30958682112, 371504185344, 4458050224128, 53496602689536, 641959232274432, 7703510787293184, 92442129447518208, 1109305553370218496, 13311666640442621952, 159739999685311463424

Offset: 0

Wolfdieter Lang, Jan 25 2002

Vincenzo Librandi, Table of n, a(n) for n = 0..900
Index entries for linear recurrences with constant coefficients, signature (12).

Cf. A067403 (third column), A067420 (fifth column), A001021 (powers of 12).

[Ceiling(6*(3*4)^(n-1)): n in [0..20]]; // Vincenzo Librandi, Oct 02 2011

Join[{1}, NestList[12*# &, 6, 20]] (* Paolo Xausa, Sep 03 2024 *)

a(n) = A067417(n+3, 3).
a(n) = 6*(3*4)^(n-1), n >= 1, a(0)=1.
G.f.: (1-6*x)/(1-12*x).
a(n) = Sum_{k=0..n} A134309(n,k)*6^k = Sum_{k=0..n} A055372(n,k)*5^k. - Philippe Deléham, Feb 04 2012

A003228 Endpoints in trees with n nodes.

Original entry on oeis.org

1, 2, 2, 5, 9, 21, 43, 101, 226, 556, 1333, 3365, 8500, 22007, 57258, 151264, 401761, 1077063, 2902599, 7871250, 21440642, 58672589, 161155637, 444240627, 1228400744, 3406668865, 9472308269, 26402207803, 73755064178

Offset: 1

N. J. A. Sloane

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. W. Robinson and A. J. Schwenk, The distribution of trees in a large random tree, Discr. Math., 12 (1975), 359-372.
Eric Weisstein's World of Mathematics, Tree Leaf.
Index entries for sequences related to trees

Cf. A000055, A003227, A055290, A055372.

a(n) = Sum_{k=1..n} k*c(n, k), where c(n, k) = A055290(n, k) has g.f. (1-x+x*y)*B(x, y)+(1/2)*(B(x^2, y^2)-B(x, y)^2) and B(x, y) is g.f. for A055372.

Corrected and extended with formula by Christian G. Bower, May 25 2000

A092811 Expansion of g.f. (1-4x)/(1-8x).

Original entry on oeis.org

1, 4, 32, 256, 2048, 16384, 131072, 1048576, 8388608, 67108864, 536870912, 4294967296, 34359738368, 274877906944, 2199023255552, 17592186044416, 140737488355328, 1125899906842624, 9007199254740992, 72057594037927936, 576460752303423488, 4611686018427387904

Offset: 0

Paul Barry, Mar 10 2004

4th binomial transform of (1,0,16,0,256,...).

Number of compositions of even natural numbers into n parts <= 7. - Adi Dani, May 28 2011

			From _Adi Dani_, May 28 2011: (Start)
a(2)=32: there are 32 compositions of even natural numbers into 2 parts <= 7:
(0,0);
(0,2),(2,0),(1,1);
(0,4),(4,0),(1,3),(3,1),(2,2);
(0,6),(6,0),(1,5),(5,1),(2,4),(4,2),(3,3);
(1,7),(7,1),(2,6),(6,2),(3,5),(5,3),(4,4);
(3,7),(7,3),(4,6),(6,4),(5,5);
(5,7),(7,5),(6,6);
(7,7).  (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (8).

Cf. A001045, A013731 (same sequence omitting initial 1), A055372, A134309.

[8^n/2+0^n/2: n in [0..20]]; // Vincenzo Librandi, Jun 16 2011

Table[EulerPhi[8^n],{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Nov 10 2009 *)

a(n)=max(1,8^n/2) \\ Charles R Greathouse IV, Apr 09 2012

def A092811(n): return 1<<3*n-1 if n else 1 # Chai Wah Wu, Apr 25 2025

a(n) = 8^n/2 + 0^n/2.
a(n) = A001045(3n+1) - A001045(3n-1) + 0^n/2.
a(n) = A013731(n-1), n > 0. - R. J. Mathar, Sep 08 2008
a(n) = 4 * 8^(n-1), a(0)=1. - Vincenzo Librandi, Jun 16 2011
a(n) = Sum_{k=0..n} A134309(n,k)*4^k = Sum_{k=0..n} A055372(n,k)*3^k. - Philippe Deléham, Feb 04 2012
E.g.f.: (1 + exp(8*x))/2. - Stefano Spezia, May 29 2024

A098402 a(n) = (0^n + 4^n * binomial(2*n,n))/2.

Original entry on oeis.org

1, 4, 48, 640, 8960, 129024, 1892352, 28114944, 421724160, 6372720640, 96865353728, 1479398129664, 22684104654848, 348986225459200, 5384358907084800, 83278084429578240, 1290810308658462720, 20045524793284362240, 311819274562201190400, 4857816066863765913600

Offset: 0

Paul Barry, Sep 06 2004

It seems that a(n) is the number of pairs of binary vectors of length 2*n which are orthogonal. (Define binary vectors here to have components of value +1 or -1. There are no pairs of binary vectors of odd length which are orthogonal.) For example, the a(1) = 4 pairs of binary vectors of length 2 are (-1,-1) and (1,-1), (-1,-1) and (-1,1), (1,-1) and (1,1), (-1,1) and (1,1). Tested up to and including a(8). - R. J. Mathar, Apr 15 2013

Tested up to and including a(210). - R. H. Hardin, May 08 2013

G. C. Greubel, Table of n, a(n) for n = 0..825

Cf. A055372, A069723, A069720, A098400, A098401.

[(0^n + 4^n*(n+1)*Catalan(n))/2: n in [0..40]]; // G. C. Greubel, Dec 27 2023

Table[(Boole[n == 0] + 4^n Binomial[2 n, n])/2, {n, 0, 18}] (* or *)
CoefficientList[Series[8 x/(# (1 - #)) &@ Sqrt[1 - 16 x], {x, 0, 18}], x] (* Michael De Vlieger, Aug 03 2016 *)

[(4^n*binomial(2*n,n) + int(n==0))/2 for n in range(41)] # G. C. Greubel, Dec 27 2023

G.f.: 8*x/( sqrt(1 - 16*x)*(1 - sqrt(1 - 16*x)) ).
a(n+1) = 4*A098400(n).
n*a(n) - 8*(2*n-1)*a(n-1) = 0. - R. J. Mathar, Nov 09 2012
a(n) ~ 16^n/(2*sqrt(Pi*n)). - Ilya Gutkovskiy, Aug 03 2016
a(n) = A055372(2*n,n). - Alois P. Heinz, Jan 21 2020
From Amiram Eldar, Jan 16 2024: (Start)
Sum_{n>=0} 1/a(n) = 17/15 + 32*arcsin(1/4)/(15*sqrt(15)).
Sum_{n>=0} (-1)^n/a(n) = 15/17 - 32*arcsinh(1/4)/(17*sqrt(17)). (End)

A362586 Triangle read by rows, T(n, k) = A094088(n) * binomial(n, k).

Original entry on oeis.org

1, 1, 1, 7, 14, 7, 121, 363, 363, 121, 3907, 15628, 23442, 15628, 3907, 202741, 1013705, 2027410, 2027410, 1013705, 202741, 15430207, 92581242, 231453105, 308604140, 231453105, 92581242, 15430207, 1619195761, 11334370327, 34003110981, 56671851635, 56671851635, 34003110981, 11334370327, 1619195761

Offset: 0

Peter Luschny, Apr 26 2023

			[0]      1;
[1]      1,       1;
[2]      7,      14,       7;
[3]    121,     363,     363,     121;
[4]   3907,   15628,   23442,   15628,    3907;
[5] 202741, 1013705, 2027410, 2027410, 1013705, 202741;

Family of triangles: A055372 (m=0, Pascal), A362585 (m=1, Fubini), this sequence (m=2, Joffe), A362849 (m=3, A278073).

Cf. A094088 (column 0 and main diagonal), A362587 (row sums).

# uses[TransOrdPart from A362585]
def A362586(n) -> list[int]: return TransOrdPart(2, n)
for n in range(6): print(A362586(n))

A362585 Triangle read by rows, T(n, k) = A000670(n) * binomial(n, k).

Original entry on oeis.org

1, 1, 1, 3, 6, 3, 13, 39, 39, 13, 75, 300, 450, 300, 75, 541, 2705, 5410, 5410, 2705, 541, 4683, 28098, 70245, 93660, 70245, 28098, 4683, 47293, 331051, 993153, 1655255, 1655255, 993153, 331051, 47293, 545835, 4366680, 15283380, 30566760, 38208450, 30566760, 15283380, 4366680, 545835

Offset: 0

Peter Luschny, Apr 26 2023

			[0]    1;
[1]    1,     1;
[2]    3,     6,     3;
[3]   13,    39,    39,    13;
[4]   75,   300,   450,   300,    75;
[5]  541,  2705,  5410,  5410,  2705,   541;
[6] 4683, 28098, 70245, 93660, 70245, 28098, 4683;

Family of triangles: A055372 (m=0, Pascal), this sequence (m=1, Fubini), A362586 (m=2, Joffe), A362849 (m=3, A278073).

Cf. A000670 (column 0 and main diagonal), A216794 (row sums).

def TransOrdPart(m, n) -> list[int]:
    @cached_function
    def P(m: int, n: int):
        R = PolynomialRing(ZZ, "x")
        if n == 0: return R(1)
        return R(sum(binomial(m * n, m * k) * P(m, n - k) * x
                 for k in range(1, n + 1)))
    T = P(m, n)
    def C(k) -> int:
        return sum(T[j] * binomial(n, k) for j in range(n + 1))
    return [C(k) for k in range(n+1)]
def A362585(n) -> list[int]: return TransOrdPart(1, n)
for n in range(6): print(A362585(n))

cp's OEIS Frontend

A081294 Expansion of (1-2*x)/(1-4*x).

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A093143 Expansion of (1-5*x)/(1-10*x).

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A081341 Expansion of exp(3*x)*cosh(3*x).

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A134309 Triangle read by rows, where row n consists of n zeros followed by 2^(n-1).

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A067419 Fourth column of triangle A067417.

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A003228 Endpoints in trees with n nodes.

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A092811 Expansion of g.f. (1-4*x)/(1-8*x).

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A081294 Expansion of (1-2x)/(1-4x).

A093143 Expansion of (1-5x)/(1-10x).

A081341 Expansion of exp(3x)cosh(3*x).

A092811 Expansion of g.f. (1-4x)/(1-8x).