A136501 -id:A136501 - cp's OEIS frontend

A136502 Matrix inverse of triangle A136501, read by rows.

1, -1, 1, 2, -2, 1, -7, 7, -4, 1, 44, -44, 26, -8, 1, -516, 516, -308, 100, -16, 1, 11622, -11622, 6959, -2296, 392, -32, 1, -512022, 512022, -306888, 101754, -17712, 1552, -64, 1, 44588536, -44588536, 26732904, -8877272, 1554404, -139104, 6176, -128, 1

Offset: 0

Paul D. Hanna, Jan 01 2008

			Triangle begins:
1;
-1, 1;
2, -2, 1;
-7, 7, -4, 1;
44, -44, 26, -8, 1;
-516, 516, -308, 100, -16, 1;
11622, -11622, 6959, -2296, 392, -32, 1;
-512022, 512022, -306888, 101754, -17712, 1552, -64, 1;
44588536, -44588536, 26732904, -8877272, 1554404, -139104, 6176, -128, 1;

Cf. A107354 (column 0), A136503 (column 2), A136504 (row sums) ; A136501 (matrix inverse).

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

G.f. for column k: 1 = Sum_{n>=0} T(n+k,k)*x^n*(1+x)^(2^(n+k)).

A121688 G.f.: Sum_{n>=0} x^n * (1+x)^(2^n).

Original entry on oeis.org

1, 2, 3, 6, 15, 49, 210, 1191, 8981, 90405, 1219297, 22105506, 540476679, 17875316557, 802011318369, 48947781204529, 4073596070782653, 463360670014324153, 72183972733773232361, 15430254274957714069057

Offset: 0

Paul D. Hanna, Aug 15 2006

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..100

Cf. A136501.

[(&+[Binomial(2^k, n-k): k in [0..n]]): n in [0..20]]; // G. C. Greubel, Mar 15 2021

A121688:= n-> add(binomial(2^k,n-k), k=0..n); seq(A121688(n), n=0..20); # G. C. Greubel, Mar 15 2021

Table[Sum[Binomial[2^k,n-k], {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Oct 05 2020 *)

PARI
```
a(n)=sum(k=0,n,binomial(2^k,n-k))
```

[sum(binomial(2^k, n-k) for k in (0..n)) for n in (0..20)] # G. C. Greubel, Mar 15 2021

a(n) = Sum_{k=0..n} C(2^k,n-k).
Lim_{n->infinity} a(n)^(1/n^2) = 2^(1/4). - Vaclav Kotesovec, Oct 05 2020
G.f.: Sum_{n>=0} ( log(1 + x)^n / n! ) / (1 - 2^n*x). - Paul D. Hanna, Jan 23 2021

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A136502 Matrix inverse of triangle A136501, read by rows.

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A121688 G.f.: Sum_{n>=0} x^n * (1+x)^(2^n).

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