A136465 -id:A136465 - cp's OEIS frontend

A136467 Triangle T, read by rows, where column 0 of T^m equals C(m*2^(n-1), n) as n=0,1,2,3,..., for all m.

1, 1, 1, 1, 4, 1, 4, 32, 16, 1, 70, 848, 576, 64, 1, 4368, 75648, 62208, 9216, 256, 1, 906192, 22313216, 21169152, 3792896, 143360, 1024, 1, 621216192, 21827627008, 23212261376, 4793434112, 223215616, 2228224, 4096, 1, 1429702652400, 71889350288384, 83889221697536, 19373156990976, 1055047811072, 13257146368, 34865152, 16384, 1, 11288510714272000, 812123016027521024, 1022353118549770240, 258404578922332160, 15923445036482560, 238096880762880, 803108552704, 549453824, 65536, 1

Offset: 0

Paul D. Hanna, Dec 31 2007

Column 0 of T^(n+1) = row n of square array A136462 defined by: A136462(n,k) = C((n+1)*2^(k-1), k);

T^n denotes the n-th matrix power of this triangle T = A136467.

			Triangle T begins:
1;
1, 1;
1, 4, 1;
4, 32, 16, 1;
70, 848, 576, 64, 1;
4368, 75648, 62208, 9216, 256, 1;
906192, 22313216, 21169152, 3792896, 143360, 1024, 1;
621216192, 21827627008, 23212261376, 4793434112, 223215616, 2228224, 4096, 1;
1429702652400, 71889350288384, 83889221697536, 19373156990976, 1055047811072, 13257146368, 34865152, 16384, 1;
11288510714272000, 812123016027521024, 1022353118549770240, 258404578922332160, 15923445036482560, 238096880762880, 803108552704, 549453824, 65536, 1; ...
Column 0 of T^m is given by: [T^m](n,0) = C(m*2^(n-1), n) for n>=0.
Matrix square T^2 begins:
1;
2, 1;
6, 8, 1;
56, 128, 32, 1;
1820, 6048, 2176, 128, 1;
201376, 912128, 419328, 34816, 512, 1;
74974368, 449708544, 249300992, 26198016, 548864, 2048, 1;
94525795200, 739136655360, 477013868544, 59943682048, 1604059136, 8650752, 8192, 1; ...
in which column 0 equals [T^2](n,0) = C(2^n, n) for n>=0.
Matrix cube T^3 begins:
1;
3, 1;
15, 12, 1;
220, 288, 48, 1;
10626, 19696, 4800, 192, 1;
1712304, 4213376, 1333504, 76800, 768, 1;
927048304, 2927926016, 1133186048, 83992576, 1216512, 3072, 1;
1708566412608, 6784661682176, 3094826778624, 278193635328, 5216272384, 19267584, 12288, 1; ...
in which column 0 equals [T^3](n,0) = C(3*2^(n-1), n) for n>=0.
Matrix 4th power T^4 begins:
1;
4, 1;
28, 16, 1;
560, 512, 64, 1;
35960, 45888, 8448, 256, 1;
7624512, 12731904, 3066880, 135168, 1024, 1;
5423611200, 11434738688, 3390050304, 193953792, 2146304, 4096, 1;
13161885792000, 34243130728448, 12032434503680, 841005662208, 12133597184, 34078720, 16384, 1; ...
in which column 0 equals [T^4](n,0) = C(4*2^(n-1), n) for n>=0.
Matrix 5th power T^5 begins:
1;
5, 1;
45, 20, 1;
1140, 800, 80, 1;
91390, 88720, 13120, 320, 1;
24040016, 30268800, 5881600, 209920, 1280, 1;
21193254160, 33353694464, 8005555200, 372858880, 3338240, 5120, 1;
63815149590720, 122539734714368, 34967493738496, 1998561607680, 23429775360, 53084160, 20480, 1; ...
in which column 0 equals [T^5](n,0) = C(5*2^(n-1), n) for n>=0.

Paul D. Hanna, Table of n, a(n) for n = 0..495

Cf. columns: A136465, A136468, A136469; A136470 (matrix square); A136462.

{T(n,k) = my(M=matrix(n+1,n+1,r,c,binomial(r*2^(c-2),c-1)),P); P=matrix(n+1,n+1,r,c,binomial((r+1)*2^(c-2),c-1));(P~*M~^-1)[n+1,k+1]}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

Diagonals: T(n+1,n) = 4^n; T(n+2,n) = (2^(n+1) + n-1)*8^n.

T(n,k) is divisible by 2^((n-k)*k) for n>=k>=0.

A136462 Square table, read by antidiagonals, where T(n,k) = C((n+1)*2^(k-1), k) for n>=0, k>=0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 6, 4, 1, 4, 15, 56, 70, 1, 5, 28, 220, 1820, 4368, 1, 6, 45, 560, 10626, 201376, 906192, 1, 7, 66, 1140, 35960, 1712304, 74974368, 621216192, 1, 8, 91, 2024, 91390, 7624512, 927048304, 94525795200, 1429702652400, 1, 9, 120, 3276, 194580, 24040016, 5423611200, 1708566412608, 409663695276000, 11288510714272000, 1, 10, 153, 4960, 367290, 61124064, 21193254160, 13161885792000, 10895665708319184, 6208116950265950720, 312268282598377321216

Offset: 0

Paul D. Hanna, Dec 31 2007

Row n equals column 0 of matrix product A136467^(n+1) for n>=0.

			1,1,1,4,70,4368,906192,621216192,1429702652400,11288510714272000,...;
1,2,6,56,1820,201376,74974368,94525795200,409663695276000,...;
1,3,15,220,10626,1712304,927048304,1708566412608,...;
1,4,28,560,35960,7624512,5423611200,13161885792000,...;
1,5,45,1140,91390,24040016,21193254160,63815149590720,...;
1,6,66,2024,194580,61124064,64300886496,231207760388736,...;
1,7,91,3276,367290,134153712,163995687856,685581099291712,...;
1,8,120,4960,635376,264566400,368532802176,1756185841659392,...; ...
Triangle A136467 begins:
1;
1,1;
1,4,1;
4,32,16,1;
70,848,576,64,1;
4368,75648,62208,9216,256,1;
906192,22313216,21169152,3792896,143360,1024,1;
621216192,21827627008,23212261376,4793434112,223215616,2228224,4096,1;
such that row n of A136462 equals column 0 of A136467^(n+1).

Paul D. Hanna, Table of n, a(n) for n = 0..495, for rows 0..30 of flattened table.

Cf. rows: A136465, A014070, A136466, A101346; A136463 (diagonal); A136467.

{T(n,k)=binomial((n+1)*2^(k-1),k)}
for(n=0,10,for(k=0,10,print1(T(n,k),", "));print(""))

/* T(n,k) = Coefficient of x^k in series: */
{T(n,k)=polcoeff(sum(i=0,k,((n+1)/2)^i*log(1+2^i*x +x*O(x^k))^i/i!),k)}
for(n=0,10,for(k=0,10,print1(T(n,k),", "));print(""))

O.g.f. of row n: Sum_{k>=0} ((n+1)/2)^k * log(1 + 2^k*x)^k / k! = Sum_{k>=0} C((n+1)*2^(k-1), k) * x^k for n>=0.

More terms and b-file added by Paul D. Hanna, Jul 02 2016

A080839 Number of positive increasing integer sequences of length n with Gilbreath transform (that is, the diagonal of leading successive absolute differences) given by {1,1,1,1,1,...}.

Original entry on oeis.org

1, 1, 1, 2, 6, 27, 180, 1786, 26094, 559127, 17535396, 804131875, 53833201737

Offset: 1

John W. Layman, Mar 28 2003

From T. D. Noe, Feb 05 2007: (Start)
The slowest-growing sequence of length n is 1,2,4,6,...,2(n-1). The fastest-growing sequence is 1,2,4,8,...,2^(n-1).
The ratio a(n+1)a(n-1)/a(n)^2 appears to converge to a constant near 1.46, which is the approximate growth rate of A001609. Are the sequences related?
(End)
Also, a(n) is the number of (not necessarily increasing) positive integer sequences of length n-1 with Gilbreath transform (1, ..., 1). - Pontus von Brömssen, May 13 2023

			The table below shows that {1,2,4,6,10} is one of the 6 sequences of length 5 that satisfy the stated condition:
   1
   2 1
   4 2 1
   6 2 0 1
  10 4 2 2 1

Cf. A001609, A036262, A363002, A363003, A363004, A363005.

Cf. also A136465, the total number of increasing sequences with the same maximum length. [From Charles R Greathouse IV, Aug 08 2010]

More terms from T. D. Noe, Feb 05 2007

Added "positive" to definition. - N. J. A. Sloane, May 13 2023

A136463 Diagonal of square array A136462: a(n) = C((n+1)*2^(n-1), n) for n>=0.

Original entry on oeis.org

1, 2, 15, 560, 91390, 61124064, 163995687856, 1756185841659392, 75079359427627897200, 12831653340946454374300160, 8777916355714456994772455584000, 24054320541767107204031746600673906688

Offset: 0

Paul D. Hanna, Dec 31 2007

nonn

a(n) is divisible by (n+1) for n>=0: a(n)/(n+1) = A136464(n).

Cf. A136462; A136464; rows: A136465, A014070, A136466, A101346; A136467.

Table[Binomial[(n+1)2^(n-1),n],{n,0,15}]  (* Harvey P. Dale, Apr 20 2011 *)

PARI
```
a(n)=binomial((n+1)*2^(n-1),n)
```

/* a(n) = Coefficient of x^n in series: */
a(n)=polcoeff(sum(i=0,n,((n+1)/2)^i*log(1+2^i*x +x*O(x^n))^i/i!),n)

a(n) = [x^n] Sum_{i>=0} ((n+1)/2)^i * log(1 + 2^i*x)^i/i!.
a(n) is found in row n, column 0, of matrix power A136467^(n+1) for n>=0.
a(n) ~ exp(n+1) * 2^(n*(n-1)) / sqrt(2*Pi*n). - Vaclav Kotesovec, Jul 02 2016

A136466 Row 2 of square array A136462: a(n) = C(3*2^(n-1), n) for n>=0.

Original entry on oeis.org

1, 3, 15, 220, 10626, 1712304, 927048304, 1708566412608, 10895665708319184, 244373929798154341120, 19561373281624772727757056, 5658395223117478029148167447552, 5975982733408602667847206514763365888

Offset: 0

Paul D. Hanna, Dec 31 2007

nonn

a(n) is found in row n, column 0, of matrix cube A136467^3 for n>=0.

Cf. A136462; other rows: A136465, A014070, A101346; A136467.

Table[Binomial[3*2^(n-1),n], {n,0,15}] (* Vaclav Kotesovec, Jul 02 2016 *)

PARI
```
a(n)=binomial(3*2^(n-1),n)
```

/* T(n,k) = Coefficient of x^k in series: */ a(n)=polcoeff(sum(i=0,n,(3/2)^i*log(1+2^i*x +x*O(x^n))^i/i!),n)

a(n) = [x^n] Sum_{i>=0} (3/2)^i * log(1 + 2^i*x)^i/i!.

a(n) ~ 3^n * 2^(n*(n-1)) / n!. - Vaclav Kotesovec, Jul 02 2016

A268408 Triangle T(d,v) read by rows: the number of hyper-tetrahedra with volume v/d! defined by selecting d+1 vertices of the d-dimensional unit-hypercube.

Original entry on oeis.org

0, 1, 0, 4, 12, 56, 2, 1360, 2672, 320, 16, 350000, 431232, 107904, 12864, 3872, 320, 255036992, 234667968, 98251776, 19523136, 10633728, 1615552, 1182720, 163520, 127360, 13440

Offset: 1

R. J. Mathar, Feb 04 2016

The unit hypercube in dimension d has 2^d vertices, conveniently expressed by their Cartesian coordinates as binary vectors of length d of 0's and 1's. Hyper-tetrahedra (simplices) are defined by selecting a subset of 1+d of them. The (signed) volume V of a tetrahedron is the determinant of the d vectors of the edges divided by d!. (The volume may be zero if some edges in the tetrahedron are linearly dependent.) The triangle T(d,v) is a histogram of all A136465(d+1) tetrahedra classified by absolute (unsigned) volume V=v/d!.

The number of non-flat simplices (row sums without the leftmost column) are tabulated by Brandts et al. (Table 1, column beta_n). - R. J. Mathar, Feb 06 2016

			In d=2, 4 tetrahedra (triangles) are defined by taking subsets of d+1=3 vertices out of the 2^2=4 vertices of the unit square. Each of them has the same volume (area) 1/2!, so T(d=2,v=1)=4.
In d=3, 12 = T(d=3,v=0) tetrahedra with zero volume are defined by taking subsets of d+1=4 vertices out of the 2^3=8 vertices of the unit cube. These are the cases of taking any 4 vertices on a common face. (There are 6 faces and two different edge sets for each of them; one with edges along the cube's edges, and one with edges along the face diagonals.)
The triangle starts in row d=1 as follows:
0 1;
0  4;
12  56  2;
1360  2672  320  16 ;
350000  431232  107904  12864  3872  320;

J. Brandts, Sander Dijkhuis et al, There are only two nonobtuse binary triangulations of the unit n-cube, arXiv:1209.3875 and Comput. Geometry 46 (2013) 286

Cf. A136465 (row sums), A003432 (maximum column index), A004145 (column v=0).

cp's OEIS Frontend

A136467 Triangle T, read by rows, where column 0 of T^m equals C(m*2^(n-1), n) as n=0,1,2,3,..., for all m.

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A136462 Square table, read by antidiagonals, where T(n,k) = C((n+1)*2^(k-1), k) for n>=0, k>=0.

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A080839 Number of positive increasing integer sequences of length n with Gilbreath transform (that is, the diagonal of leading successive absolute differences) given by {1,1,1,1,1,...}.

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A136463 Diagonal of square array A136462: a(n) = C((n+1)*2^(n-1), n) for n>=0.

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A136466 Row 2 of square array A136462: a(n) = C(3*2^(n-1), n) for n>=0.

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A268408 Triangle T(d,v) read by rows: the number of hyper-tetrahedra with volume v/d! defined by selecting d+1 vertices of the d-dimensional unit-hypercube.

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