A055372 -id:A055372 - cp's OEIS frontend

A362849 Triangle read by rows, T(n, k) = A243664(n) * binomial(n, k).

1, 1, 1, 21, 42, 21, 1849, 5547, 5547, 1849, 426405, 1705620, 2558430, 1705620, 426405, 203374081, 1016870405, 2033740810, 2033740810, 1016870405, 203374081, 173959321557, 1043755929342, 2609389823355, 3479186431140, 2609389823355, 1043755929342, 173959321557

Offset: 0

Peter Luschny, May 05 2023

			[0]         1;
[1]         1,          1;
[2]        21,         42,         21;
[3]      1849,       5547,       5547,       1849;
[4]    426405,    1705620,    2558430,    1705620,     426405;
[5] 203374081, 1016870405, 2033740810, 2033740810, 1016870405, 203374081;

Family of triangles: A055372 (m=0, Pascal), A362585 (m=1, Fubini), A362586 (m=2, Joffe), this sequence (m=3, A278073).

Cf. A243664 (column 0 and main diagonal).

# uses[TransOrdPart from A362585]
def A362849(n) -> list[int]: return TransOrdPart(3, n)
for n in range(6): print(A362849(n))

A055373 Invert transform applied twice to Pascal's triangle A007318.

Original entry on oeis.org

1, 1, 1, 3, 6, 3, 9, 27, 27, 9, 27, 108, 162, 108, 27, 81, 405, 810, 810, 405, 81, 243, 1458, 3645, 4860, 3645, 1458, 243, 729, 5103, 15309, 25515, 25515, 15309, 5103, 729, 2187, 17496, 61236, 122472, 153090, 122472, 61236, 17496, 2187, 6561

Offset: 0

Christian G. Bower, May 16 2000

Triangle T(n,k), 0 <= k <= n, read by rows, given by [1, 2, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 2, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 10 2005

T(n,k) is the number of sequences of nonempty sequences of nonempty bit strings with n bits and exactly k 1's over all strings in the sequence of sequences. In other words, these are sequences of the structures counted by A055372. - Geoffrey Critzer, Apr 06 2013

			Triangle begins:
   1;
   1,   1;
   3,   6,   3;
   9,  27,  27,   9;
  27, 108, 162, 108,  27;
  ...

N. J. A. Sloane, Transforms
Index entries for triangles and arrays related to Pascal's triangle

Cf. A000244, A007318, A055372, A055374.

Cf. A084938.

nn=10;f[list_]:=Select[list,#>0&];a=(x+y x)/(1-(x+y x));b=1/(1-a);Map[f,CoefficientList[Series[1/(2-b),{x,0,nn}],{x,y}]]//Grid  (* Geoffrey Critzer, Apr 06 2013 *)

T(n,k) = 3^(n-1)*C(n, k) for n > 0.

O.g.f.: 1/(2 - A(x,y)) where A(x,y) is the o.g.f. for A055372. - Geoffrey Critzer, Apr 06 2013

A055374 Invert transform applied three times to Pascal's triangle A007318.

Original entry on oeis.org

1, 1, 1, 4, 8, 4, 16, 48, 48, 16, 64, 256, 384, 256, 64, 256, 1280, 2560, 2560, 1280, 256, 1024, 6144, 15360, 20480, 15360, 6144, 1024, 4096, 28672, 86016, 143360, 143360, 86016, 28672, 4096, 16384, 131072, 458752, 917504, 1146880, 917504

Offset: 0

Christian G. Bower, May 16 2000

Triangle T(n,k), 0 <= k <= n, read by rows, given by [1, 3, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 3, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 10 2005

			   1;
   1,   1;
   4,   8,   4;
  16,  48,  48,  16;
  64, 256, 384, 256, 64; ...

N. J. A. Sloane, Transforms
Index entries for triangles and arrays related to Pascal's triangle

Cf. A000302, A007318, A055372, A055373.

a(n,k) = 4^(n-1)*C(n, k), for n>0.

G.f.: (-1+3*x+3*x*y)/(-1+4*x+4*x*y). - R. J. Mathar, Aug 12 2015

A143376 Triangle read by rows: T(n,k) is the number of unordered pairs of vertices at distance k in the cube Q_n of dimension n (1 <= k <= n).

Original entry on oeis.org

1, 4, 2, 12, 12, 4, 32, 48, 32, 8, 80, 160, 160, 80, 16, 192, 480, 640, 480, 192, 32, 448, 1344, 2240, 2240, 1344, 448, 64, 1024, 3584, 7168, 8960, 7168, 3584, 1024, 128, 2304, 9216, 21504, 32256, 32256, 21504, 9216, 2304, 256, 5120, 23040, 61440, 107520

Offset: 1

Emeric Deutsch, Sep 05 2008

Sum of entries in row n = 2^(n-1)*(2^n-1) = A006516.
The entries in row n are the coefficients of the Wiener polynomial of the cube Q_n.
Sum_{k=1..n} k*T(n,k) = n*4^(n-1) = A002697(n) = the Wiener index of the cube Q_n.
Triangle T(n,k), 1 <= k <= n, read by rows given by [1,1,0,0,0,0,0,...]DELTA[1,1,0,0,0,0,0,...] where DELTA is the operator defined in A084938; subtriangle of triangle A055372. - Philippe Deléham, Oct 14 2008

			T(2,1)=4, T(2,2)=2 because in Q_2 (a square) there are 4 distances equal to 1 and 2 distances equal to 2.
Triangle starts:
   1;
   4,   2;
  12,  12,   4;
  32,  48,  32,   8;
  80, 160, 160,  80,  16;

Indranil Ghosh, Rows 1..125, flattened
B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969.

Cf. A002697, A006516.

T:=proc(n,k) options operator, arrow: 2^(n-1)*binomial(n,k) end proc: for n to 10 do seq(T(n,k),k=1..n) end do; # yields sequence in triangular form

nn = 8; A[u_, z_] := (z + u z)/(1 - (z + u z));
Drop[Map[Select[#, # > 0 &] &, Map[Drop[#, 1] &,CoefficientList[Series[1/(1 - A[u, z]), {z, 0, nn}], {z, u}]]],1] // Grid (* Geoffrey Critzer, Mar 04 2017 *)
Flatten[Table[2^(n-1) Binomial[n, k], {n, 10},{k,n}]] (* Indranil Ghosh, Mar 06 2017 *)

tabl(nn) = {for (n=1, nn, for(k=1, n, print1(2^(n-1) * binomial(n, k),", ");); print();); };
tabl(10); \\ Indranil Ghosh, Mar 06 2017

import math
f=math.factorial
def C(n,r): return f(n) / f(r) / f(n-r)
i=1
for n in range(1,126):
    for k in range(1,n+1):
        print(str(i)+" "+str(2**(n-1)*C(n,k)))
        i+=1 # Indranil Ghosh, Mar 06 2017

T(n,k) = 2^(n-1)*binomial(n,k).
G.f.: G(q,z) = qz/((1-2z)(1-2z-2zq)).
T(n,k) = A055372(n,k). - Philippe Deléham, Oct 14 2008

Typo corrected by Philippe Deléham, Jan 05 2009

cp's OEIS Frontend

A362849 Triangle read by rows, T(n, k) = A243664(n) * binomial(n, k).

Views

Author

Keywords

Examples

Crossrefs

Programs

SageMath

A055373 Invert transform applied twice to Pascal's triangle A007318.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A055374 Invert transform applied three times to Pascal's triangle A007318.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A143376 Triangle read by rows: T(n,k) is the number of unordered pairs of vertices at distance k in the cube Q_n of dimension n (1 <= k <= n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

Extensions