A136467 -id:A136467 - cp's OEIS frontend

A136468 Column 1 of triangle A136467 (scaled): a(n) = A136467(n+1,1)/2^n.

1, 2, 8, 106, 4728, 697288, 341056672, 561635549128, 3172355531357504, 62573893999774791616, 4377792727679018712191744, 1100465546170585425117622597248, 1004426091768772936789017838438890496

Offset: 0

Paul D. Hanna, Dec 31 2007

nonn

Cf. A136467; A136469.

{a(n)=local(M=matrix(n+2,n+2,r,c,binomial(r*2^(c-2),c-1)),P); P=matrix(n+2,n+2,r,c,binomial((r+1)*2^(c-2),c-1));(P~*M~^-1)[n+2,2]/2^n}

A136469 Column 2 of triangle A136467 (scaled): a(n) = A136467(n+2,2)/4^n.

Original entry on oeis.org

1, 4, 36, 972, 82692, 22668224, 20480767016, 62399482333360, 654732295896172624, 24092174555680443592896, 3156168229958886081384337440, 1490220874303979634022445823087616

Offset: 0

Paul D. Hanna, Dec 31 2007

nonn

Cf. A136467; A136468.

{a(n)=local(M=matrix(n+3,n+3,r,c,binomial(r*2^(c-2),c-1)),P); P=matrix(n+3,n+3,r,c,binomial((r+1)*2^(c-2),c-1));(P~*M~^-1)[n+3,3]/4^n}

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

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

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).

A179430 Triangular matrix T where column 0 of T^m equals C(m*3^(n-1), n) at row n for n>=0, m>=0.

Original entry on oeis.org

1, 1, 1, 3, 9, 1, 84, 405, 81, 1, 17550, 121500, 32805, 729, 1, 25621596, 247203171, 82255257, 2539107, 6561, 1, 268715232324, 3543210805275, 1382411964132, 53628242751, 199290375, 59049, 1, 21091830512086620, 373203783345533355

Offset: 0

Paul D. Hanna, Jul 20 2010

			Triangle T begins:
1;
1, 1;
3, 9, 1;
84, 405, 81, 1;
17550, 121500, 32805, 729, 1;
25621596, 247203171, 82255257, 2539107, 6561, 1;
268715232324, 3543210805275, 1382411964132, 53628242751, 199290375, 59049, 1;
21091830512086620, 373203783345533355, 165018275857291311, 7607829219099993, 36456526295226, 15884240049, 531441, 1; ...
where column 0 of T equals A179431(n) = C(3^(n-1), n):
[1, 1, 3, 84, 17550, 25621596, 268715232324, ...]. ...
Illustrate row n in column 0 of T^m equals C(m*3^(n-1), n) as follows.
Matrix square T^2 begins:
1;
2, 1;
15, 18, 1;
816, 1539, 162, 1;
316251, 833490, 124659, 1458, 1;
873642672, 3060203490, 585411786, 9861183, 13122, 1; ...
where column 0 of T^2 equals A179432(n) = C(2*3^(n-1), n):
[1, 2, 15, 816, 316251, 873642672, 17743125256857, ...]. ...
Matrix cube T^3 begins:
1;
3, 1;
36, 27, 1;
2925, 3402, 243, 1;
1663740, 2667411, 275562, 2187, 1;
6774333588, 14164214850, 1896890076, 21966228, 19683, 1; ...
where column 0 of T^3 equals A136393(n) = C(3^n, n):
[1, 3, 36, 2925, 1663740, 6774333588, 204208594169580, ...].

Cf. A179431, A179432, A136393, A179433, A179434, variant: A136467.

{T(n, k)=local(M=matrix(n+1, n+1, r, c, binomial(r*3^(c-2), c-1)), P); P=matrix(n+1, n+1, r, c, binomial((r+1)*3^(c-2), c-1)); (P~*M~^-1)[n+1, k+1]}

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

cp's OEIS Frontend

A136468 Column 1 of triangle A136467 (scaled): a(n) = A136467(n+1,1)/2^n.

Views

Author

Keywords

Crossrefs

Programs

PARI

A136469 Column 2 of triangle A136467 (scaled): a(n) = A136467(n+2,2)/4^n.

Views

Author

Keywords

Crossrefs

Programs

PARI

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

A179430 Triangular matrix T where column 0 of T^m equals C(m*3^(n-1), n) at row n for n>=0, m>=0.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI