A126150 -id:A126150 - cp's OEIS frontend

A126152 Main diagonal of symmetric triangle A126150: a(n) = A126150(n,n).

1, 4, 36, 744, 28536, 1736064, 152914176, 18372559104, 2885671339776, 573765893121024, 140835811776316416, 41820352964911908864, 14774712204104658671616, 6124078747943873540112384

Offset: 0

Paul D. Hanna, Dec 19 2006

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..232

Cf. A126150; A126151 (column 0), A126153 (diagonal).

/* Continued fraction involving even-indexed pentagonal numbers: */
{a(n)=local(CF=1+x*O(x),m,P); for(k=1, n,m=2*((n-k)\2+1);P=m*(3*m-1)/2-((n-k+1)%2); CF=1/(1-P*x*CF)); polcoeff(CF, n, x)}
for(n=0,20,print1(a(n),","))

a(n) = Sum_{k=0..n} A130847(n,k)*3^k. - Philippe Deléham, Jul 22 2007
G.f.: 1/(1 - 4*x/(1 - 5*x/(1 - 21*x/(1 - 22*x/(1 - 50*x/(1 - 51*x/(1 - 91*x/(1 - 92*x/(1 -...))))))))), a continued fraction involving even-indexed pentagonal numbers A000326. - Paul D. Hanna, Feb 15 2012
a(n) ~ Gamma(1/3) * 2^(3*n+7/3) * 3^(n+3/2) * n^(2*n+7/6) / (exp(2*n) * Pi^(2*n+13/6)). - Vaclav Kotesovec, May 30 2015

A126153 Secondary diagonal of symmetric triangle A126150: a(n) = A126150(n+1,n).

Original entry on oeis.org

1, 24, 636, 26304, 1650456, 147705984, 17913816576, 2830553662464, 565108879101696, 139114514096953344, 41397845529582959616, 14649251145209922945024, 6079754611331559564097536

Offset: 0

Paul D. Hanna, Dec 19 2006

nonn

Cf. A126150; A126151 (column 0), A126152 (diagonal).

a(n) = A126152(n+1) - 3*A126152(n), where A126152 is the main diagonal of triangle A126150.

A126154 a(n) = Sum_{k=0..n} C(2n,k)*A126150(n,k).

Original entry on oeis.org

1, 10, 420, 38760, 6274320, 1574566560, 566087814720, 276000683679360, 175277074975269120, 140576092830146972160, 138951226585459200107520, 165951785120565606919464960, 235593066998837613695124049920

Offset: 0

Paul D. Hanna, Dec 20 2006

nonn

			a(1) = 1*(1) + 2*(4) + 1*(1) = 10;
a(2) = 1*(6) + 4*(24) + 6*(36) + 4*(24) + 1*(6) = 120;
a(3) = 1*(96) + 6*(384) + 15*(636) + 20*(744) + 15*(636) + 6*(384) + 1*(96) = 38760.

Cf. A126150.

A126151 E.g.f.: ( (1 + cos(sqrt(6)*x))/2 )^(-1/3), showing coefficients of only the even powers of x.

Original entry on oeis.org

1, 1, 6, 96, 2976, 151416, 11449296, 1204566336, 168233625216, 30110372009856, 6719377991060736, 1829013279998846976, 596449130341224185856, 229556544889929225117696, 102956750031135241952280576, 53228316147100497514507862016, 31423560379886826670772937424896

Offset: 0

Paul D. Hanna, Dec 19 2006

nonn

Previous name was: Column 0 and row sums of symmetric triangle A126150.

			E.g.f.: A(x) = 1 + x^2/2! + 6*x^4/4! + 96*x^6/6! + 2976*x^8/8! + 151416*x^10/10! +...
where the logarithm begins:
log(A(x)) = x^2/2! + 3*x^4/4! + 36*x^6/6! + 918*x^8/8! + 40176*x^10/10! + 2686608*x^12/12! +...
compare the logarithm to
A(x)^3 = 1 + 3*x^2/! + 36*x^4/4! + 918*x^6/6! + 40176*x^8/8! + 2686608*x^10/10! +...
where A(x)^3 = 2/(1 + cos(sqrt(6)*x)).

Vaclav Kotesovec, Table of n, a(n) for n = 0..232
E. Norton, Symplectic Reflection Algebras in Positive Characteristic as Ore Extensions, arXiv preprint arXiv:1302.5411 [math.RA], 2013.

Cf. A126150; diagonals: A126152, A126153.

Cf. A000326.

A000326 := n -> n * (3 * n - 1) / 2;
T := proc(n, k) option remember; if k = 0 then 1 else if k = n then T(n, k-1) else A000326(n - k + 1) * T(n, k - 1) + T(n - 1, k) fi fi end:
a := n -> T(n, n): seq(a(n), n = 0..16);  # Peter Luschny, Sep 30 2023

terms = 18;
CoefficientList[((1 + Cos[Sqrt[6] x])/2)^(-3^(-1)) + O[x]^(2 terms), x] Range[0, 2 terms - 2]! // DeleteCases[#, 0]& (* Jean-François Alcover, Jul 26 2018 *)

/* Continued Fraction involving pentagonal numbers A000326: */
{a(n)=local(CF=1+x*O(x),m,P); for(k=1, n,m=n-k+1;P=m*(3*m-1)/2; CF=1/(1-P*x*CF)); polcoeff(CF, n, x)}
for(n=0,20,print1(a(n),","))

/* E.g.f. A(x) = exp( Integral^2 A(x)^3 dx^2 ): */
{a(n)=local(A=1+x*O(x)); for(i=1, n, A=exp(intformal(intformal(A^3 + x*O(x^(2*n))))) ); (2*n)!*polcoeff(A, 2*n, x)}
for(n=0,20,print1(a(n),", "))

/* E.g.f. A(x) = exp( Integral A(x)^(3/2) * Integral 1/A(x)^(3/2) dx dx ) */
{a(n) = local(A=1+x); for(i=1,n, A = exp( intformal( A^(3/2) * intformal( 1/A^(3/2) + x*O(x^n)) ) ) ); n!*polcoeff(A,n)}
for(n=0,20,print1(a(2*n),", "))

a(n) = Sum_{0<=k<=n} A087736(n,k)*2^(n-k). - Philippe Deléham, Jul 17 2007
G.f.: 1/(1-x/(1-5*x/(1-12*x/(1-22*x/(1-35*x/(1-51*x/(1-70*x/(1-...- (n*(3*n-1)/2)*x/(1-...))))))))), a continued fraction involving pentagonal numbers A000326. - Paul D. Hanna, Feb 15 2012
E.g.f. satisfies: A(x) = exp( Integral Integral A(x)^3 dx dx ), where A(x) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)! and the constant of integration is zero. - Paul D. Hanna, May 29 2015
E.g.f. satisfies: A(x) = exp( Integral A(x)^(3/2) * Integral 1/A(x)^(3/2) dx dx ),  where A(x) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)! and the constant of integration is zero. - Paul D. Hanna, Jun 02 2015
a(n) ~ Gamma(1/3) * 2^(3*n+4/3) * 3^(n+1/2) * n^(2*n+1/6) / (exp(2*n) * Pi^(2*n+7/6)). - Vaclav Kotesovec, May 30 2015
The computation can be based on the pentagonal numbers, a(n) = T(n, n) where T(n, k) = A000326(n - k + 1) * T(n, k - 1) + T(n - 1, k) for 0 < k < n, and T(n, 0) = 1, T(n, n) = T(n, n-1) if n > 0. This is equivalent to Paul D. Hanna's continued fraction 2012. - Peter Luschny, Sep 30 2023

New name from Paul D. Hanna, May 30 2015

A126155 Symmetric triangle, read by rows of 2*n+1 terms, similar to triangle A008301. Second term 5 times first term.

Original entry on oeis.org

1, 1, 5, 1, 7, 35, 55, 35, 7, 139, 695, 1195, 1415, 1195, 695, 139, 5473, 27365, 48145, 63365, 69025, 63365, 48145, 27365, 5473, 357721, 1788605, 3175705, 4343885, 5126905, 5403005, 5126905, 4343885, 3175705, 1788605, 357721

Offset: 0

Paul D. Hanna, Dec 20 2006

			The triangle begins:
  1;
  1, 5, 1;
  7, 35, 55, 35, 7;
  139, 695, 1195, 1415, 1195, 695, 139;
  5473, 27365, 48145, 63365, 69025, 63365, 48145, 27365, 5473;
  357721, 1788605, 3175705, 4343885, 5126905, 5403005, 5126905, 4343885, 3175705, 1788605, 357721; ...
If we write the triangle like this:
  .......................... ....1;
  ................... ....1, ....5, ....1;
  ............ ....7, ...35, ...55, ...35, ....7;
  ..... ..139, ..695, .1195, .1415, .1195, ..695, ..139;
  .5473, 27365, 48145, 63365, 69025, 63365, 48145, 27365, .5473;
then the first term in each row is the sum of the previous row:
  5473 = 139 + 695 + 1195 + 1415 + 1195 + 695 + 139
the next term is 5 times the first:
  27365 = 5*5473,
and the remaining terms in each row are obtained by the rule illustrated by:
  48145 = 2*27365 - 5473 - 8*139;
  63365 = 2*48145 - 27365 - 8*695;
  69025 = 2*63365 - 48145 - 8*1195;
  63365 = 2*69025 - 63365 - 8*1415;
  48145 = 2*63365 - 69025 - 8*1195;
  27365 = 2*48145 - 63365 - 8*695;
  5473 = 2*27365 - 48145 - 8*139.
An alternate recurrence is illustrated by:
  27365 = 5473 + 4*(139 + 695 + 1195 + 1415 + 1195 + 695 + 139);
  48145 = 27365 + 4*(695 + 1195 + 1415 + 1195 + 695);
  63365 = 48145 + 4*(1195 + 1415 + 1195);
  69025 = 63365 + 4*(1415);
and then for k>n, T(n,k) = T(n,2n-k).

Cf. A126156 (column 0); diagonals: A126157, A126158; A126159; variants: A008301 (p=1), A125053 (p=2), A126150 (p=3).

T := proc(n,k) option remember; local j;
  if n = 1 then 1
elif k = 1 then add(T(n-1, j), j=1..2*n-3)
elif k = 2 then 5*T(n, 1)
elif k > n then T(n, 2*n-k)
else 2*T(n, k-1)-T(n, k-2)-8*T(n-1, k-2)
  fi end:
seq(print(seq(T(n,k), k=1..2*n-1)), n=1..6); # Peter Luschny, May 12 2014

T[n_, k_] := T[n, k] = Which[n==1, 1, k==1, Sum[T[n-1, j], {j, 1, 2n-3}], k==2, 5 T[n, 1], k>n, T[n, 2n-k], True, 2 T[n, k-1] - T[n, k-2] - 8 T[n-1, k-2]];
Table[T[n, k], {n, 1, 6}, {k, 1, 2n-1}] (* Jean-François Alcover, Jun 15 2019, from Maple *)

PARI
```
{T(n,k) = local(p=4);if(2*n
				
```

/* Alternate Recurrence: */
{T(n,k) = local(p=4);if(2*n

from functools import cache
@cache
def R(n, k):
      return (1 if n == 1 else sum(R(n-1, j) for j in range(1, 2*n-2))
                if k == 1 else 5*R(n, 1) if k == 2 else R(n, 2*n-k)
                if k > n else 2*R(n, k-1) - R(n, k-2) - 8*R(n-1, k-2))
def A126155(n, k): return R(n+1, k+1)
for n in range(5): print([A126155(n, k) for k in range(2*n+1)])
# Peter Luschny, Dec 14 2023

Sum_{k=0..2n} (-1)^k*C(2n,k)*T(n,k) = (-8)^n.

cp's OEIS Frontend

A126152 Main diagonal of symmetric triangle A126150: a(n) = A126150(n,n).

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A126153 Secondary diagonal of symmetric triangle A126150: a(n) = A126150(n+1,n).

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A126154 a(n) = Sum_{k=0..n} C(2n,k)*A126150(n,k).

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A126151 E.g.f.: ( (1 + cos(sqrt(6)*x))/2 )^(-1/3), showing coefficients of only the even powers of x.

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A126155 Symmetric triangle, read by rows of 2*n+1 terms, similar to triangle A008301. Second term 5 times first term.

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