A136462 -id:A136462 - cp's OEIS frontend

A136465 Row 0 of square array A136462: a(n) = C(2^(n-1), n) for n>=0.

1, 1, 1, 4, 70, 4368, 906192, 621216192, 1429702652400, 11288510714272000, 312268282598377321216, 30813235422145714150738944, 11005261717918037175659349191168, 14391972654784168932973746746691440640, 69538271351155829150354851003285125277716480, 1250303357941919088313448625534941836891635347865600

Offset: 0

Paul D. Hanna, Dec 31 2007

a(n) is found in row n, column 0, of triangle A136467 for n>=0.
For n > 0, number of increasing integer sequences 1 <= a_1 < ... < a_n <= 2^(n-1). - Charles R Greathouse IV, Aug 08 2010
The (n-1)-dimensional hypercube has 2^(n-1) corners.  There are binomial(2^(n-1),n) ways of selecting a set of n corners. So a(n) is the number of simplices (hyper-tetrahedra) with vertices defined by a corner subset of a (n-1)-dimensional hypercube. (This count includes degenerate polytopes with zero volume.) - R. J. Mathar, Jan 16 2016

			From _Paul D. Hanna_, Sep 26 2010: (Start)
G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 70*x^4 + 4368*x^5 +...
A(x) = 1 + log(1+2*x)/2 + log(1 + 2^2*x)^2/(2!*2^2) + log(1 + 2^3*x)^3/(3!*2^3) + log(1 + 2^4*x)^4/(4!*2^4) +... (End)

J. Brandts, S. Dijkhuis, V. de Haan, M. Krizek, There are only two nonobtuse binary triangulations of the unit n-cube, Comp. Geom. 46 (2013) 286-297, Table 1.

Cf. A136462; other rows: A014070, A136466, A101346; A136467.

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

{a(n)=binomial(2^(n-1),n)}
for(n=0,20,print1(a(n),", "))

/* a(n) = Coefficient of x^k in series: */
{a(n)=polcoeff(sum(i=0,n,(1/2)^i*log(1+2^i*x +x*O(x^n))^i/i!),n)}
for(n=0,20,print1(a(n),", "))

{a(n)=polcoeff(sum(m=0,n,log(1+2^m*x+x*O(x^n))^m/(m!*2^m)),n)}
for(n=0,20,print1(a(n),", ")) \\ Paul D. Hanna, Sep 26 2010

a(n) = [x^n] Sum_{i>=0} (1/2)^i * log(1 + 2^i*x)^i/i!.
O.g.f.: Sum_{n>=0} log(1 + 2^n*x)^n / (n!*2^n). - Paul D. Hanna, Sep 26 2010
a(n) ~ 2^(n*(n-1)) / n!. - Vaclav Kotesovec, Jul 02 2016

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

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.

Original entry on oeis.org

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.

A136470 Triangle T, read by rows, where column 0 of T^m = {C(m*2^n, n), n>=0} for all m.

Original entry on oeis.org

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, 409663695276000, 4132411271661568, 3028532448264192, 439222754869248, 14159357935616, 98723430400, 136839168, 32768, 1, 6208116950265950720, 80121787455478857728, 65415571433959456768, 10679727629898088448, 399723620798038016, 3391703461396480, 6141702569984, 2172649472, 131072, 1

Offset: 0

Paul D. Hanna, Dec 31 2007

Column 0 of T^(n+1) = row 2n+1 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 = A136470.

			Triangle T 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;
409663695276000, 4132411271661568, 3028532448264192, 439222754869248, 14159357935616, 98723430400, 136839168, 32768, 1; ...
Column 0 of T^m is given by: [T^m](n,0) = C(m*2^n, n) for n>=0.

Cf. columns: A014070, A136471, A136472; A136467 (matrix square-root); A136462.

{T(n,k)=local(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)^2)[n+1,k+1]}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))

Equals the matrix square of triangle A136467.

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

A136464 C((n+1)*2^(n-1),n)/(n+1).

Original entry on oeis.org

1, 1, 5, 140, 18278, 10187344, 23427955408, 219523230207424, 8342151047514210800, 1283165334094645437430016, 797992395974041544979314144000, 2004526711813925600335978883389492224, 20319947544434703948516669537695298775460352

Offset: 0

Paul D. Hanna, Dec 31 2007

Cf. A136463, A136462.

[1] cat [(Binomial((n+1)*2^(n-1), n))/(n+1): n in [1..15]]; // Vincenzo Librandi, Nov 20 2014

A136464:=n->binomial((n+1)*2^(n-1),n)/(n+1): seq(A136464(n), n=0..15); # Wesley Ivan Hurt, Nov 19 2014

Table[Binomial[(n + 1)*2^(n - 1), n]/(n + 1), {n, 0, 15}] (* Wesley Ivan Hurt, Nov 19 2014 *)

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

a(n) = A136463(n)/(n+1).

a(12) from Vincenzo Librandi, Nov 20 2014

cp's OEIS Frontend

A136465 Row 0 of square array A136462: a(n) = C(2^(n-1), n) for n>=0.

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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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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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A136470 Triangle T, read by rows, where column 0 of T^m = {C(m*2^n, n), n>=0} for all m.

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A136464 C((n+1)*2^(n-1),n)/(n+1).

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