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# User talk:Peter Bala

Peter:

```    I'm interested in your addition to A036969, on the central factorial numbers.  Specifically,
```

what's the source for your comment that they're the connection coefficients in the expansion x^n = \prod T(n,k) (x-k^2)? This is quite correct, I'm just trying to source it. It's different from the Riordan expansion.

Thanks, William Keith P.S. You can reply at william.keith@gmail.com if it's more convenient. And, of course, you can delete this note once you've read it. It's your page.  :^) William J. Keith 19:30, 31 January 2012 (UTC)

Peter, I just wanted to say I'm honored to have my work referenced in your paper at http://oeis.org/A220335/a220335.pdf . It makes me feel that it wasn't a complete waste of time :) Drop me a line some-time.. stephen.crowley@hushmail.com

```Couple questions
```

Hi Peter,

Re your "pink comment" in A060294, particularly - "It is one of Ramanujan's - see Zudilin arXiv: 0712.1332v2 equation (1.3) for a reference." Does Zudilin's list the formula in that paper of his in generalized form, formulated by Paul Hanna that is as 2/Pi = Sum_{n>0} (-1)^n * (4*n+1) * Product_{k=1..n} (2*k-1)^3/(2*k)^3. which BTW in Maple format is 2/Pi = sum((-1)^n * (4*n+1) * product((2*k-1)^3/(2*k)^3,k=1..n),n=0...infinity) or does it list it the way Earls put it (with Paul Hanna's correction of course) that is as 2/Pi = 1 - 5*(1/2)^3 + 9*((1*3)/(2*4))^3 - 13*((1*3*5)/(2*4*6))^3 ... ?

By now (after approval) the formula section still misses generalized form

2/Pi = Sum_{n>=0} (-1)^n * (4*n+1) * Product_{k=1..n} (2*k-1)^3/(2*k)^3.

Paul only noted it in the pink section

I think that someone should add it there still - with appropriate attribution - could you do that ?

Also since we are on Pi formula subject - I have another question to ask. Does Zudilin's paper list both a) and b) (or one - then which ?) versions of 24/Pi formulas referenced in A132714, A220852 and in A220853 as:

a) 24/Pi = sum(k>=0, (30*k+7)*C(2*k,k)^2*(Hypergeometric2F1[1/2 - k/2, -k/2, 1, 64])/(-256)^k ). or in Maple style format sum((30*k+7) * binom(2k,k)^2 * (Hypergeometric2F1[1/2 - k/2, -k/2, 1,64])/(-256)^k, k=0...infinity)

Another version of this identity is: b) 24/Pi = sum_{k>=0} ((30*k+7) * binomial(2k,k)^2 * (sum_{m=0,k/2} (binomial(k-m,m) * binomial(k,m) * 16^m))/(-256)^k)

or in Mathematica style format 24/Pi = Sum[(30*k+7) * Binomial[2k,k]^2 * (Sum[Binomial[k-m,m] * Binomial[k,m] * 16^m, {m,0,k/2}])/(256)^k, {k,0,infinity}].

??????

Cheers, Alexander R. Povolotsky 22:14, 24 March 2013 (UTC)

## Maple code in A008287

Hi Peter

In Sep 07 2013 you added a formula T(n,k) = sum... and a corresponding Maple program to compute r-nomials.

We recently discovered the same nice formula and spent some time in a quite short proof, but we didn't find any papers containing this formula. Your notice on this site is the only one we have seen so far. Do you have a reference to a corresponding paper? It seems to us that several communities (gray-codes, integer compositions, ...) are not aware that there exists such a simple formula.

Christoph Christoph Stamm 12:35, 11 June 2014 (UTC)