A136466 -id:A136466 - cp's OEIS frontend

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

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

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

Original entry on oeis.org

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

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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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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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