A300279 -id:A300279 - cp's OEIS frontend

A300050 G.f. A(x) satisfies: A(x) = Sum_{n>=0} (1 + x*A(x)^n)^n / 2^(n+1).

1, 1, 4, 28, 250, 2558, 28594, 340486, 4255344, 55300776, 742732646, 10267434138, 145692400018, 2118364506746, 31529605958892, 480186833802260, 7483472464151002, 119397596900634238, 1951747376021480874, 32721008993895160926, 563260078608381337148, 9967709437187109520736, 181544883799333028286098, 3406426523158387599683478, 65894531117591548919270114

Offset: 0

Paul D. Hanna, Feb 27 2018

nonn

Compare to: G(x) = Sum_{n>=0} (1 + x*G(x)^k)^n / 2^(n+1) holds when G(x) = 1 + x*G(x)^(k+1) for fixed k.

			G.f.: A(x) = 1 + x + 4*x^2 + 28*x^3 + 250*x^4 + 2558*x^5 + 28594*x^6 + 340486*x^7 + 4255344*x^8 + 55300776*x^9 + 742732646*x^10 + ...
such that
A(x) = 1/2 + (1 + x*A(x))/2^2 + (1 + x*A(x)^2)^2/2^3 + (1 + x*A(x)^3)^3/2^4 + (1 + x*A(x)^4)^4/2^5 + (1 + x*A(x)^5)^5/2^6 + (1 + x*A(x)^6)^6/2^7 + ...
Also, due to a series identity,
A(x) = 1 + x*A(x)/(2 - A(x))^2 + x^2*A(x)^4/(2 - A(x)^2)^3 + x^3*A(x)^9/(2 - A(x)^3)^4 + x^4*A(x)^16/(2 - A(x)^4)^5 + x^5*A(x)^25/(2 - A(x)^5)^6 + x^6*A(x)^36/(2 - A(x)^6)^7 + ... + x^n * A(x)^(n^2) / (2 - A(x)^n)^(n+1) + ...

Paul D. Hanna, Table of n, a(n) for n = 0..100

Cf. A302103, A302104, A302105, A300279.

{a(n) = my(A=1); for(i=0,n, A = sum(m=0,n, x^m * A^(m^2) / (2 - A^m + x*O(x^n))^(m+1) )); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

G.f. A(x) satisfies:
(1) A(x) = Sum_{n>=0} (1 + x*A(x)^n)^n / 2^(n+1).
(2) A(x) = Sum_{n>=0} x^n * A(x)^(n^2) / (2 - A(x)^n)^(n+1).

A302765 Decimal expansion of constant: B = Sum_{n>=1} (-1)^(n-1) / (2^n - 1)^n.

Original entry on oeis.org

8, 9, 1, 7, 8, 4, 6, 2, 2, 6, 1, 0, 9, 5, 3, 3, 4, 9, 7, 1, 5, 8, 9, 0, 1, 3, 6, 0, 6, 0, 2, 3, 9, 4, 2, 1, 0, 2, 2, 2, 1, 6, 9, 7, 0, 3, 6, 6, 1, 3, 9, 1, 8, 9, 3, 3, 6, 8, 2, 2, 3, 6, 0, 1, 2, 7, 6, 1, 2, 2, 3, 7, 8, 1, 7, 5, 4, 4, 4, 5, 5, 8, 3, 9, 6, 7, 8, 6, 4, 6, 3, 8, 6, 1, 7, 6, 3, 7, 1, 0, 5, 7, 4, 3, 9, 0, 9, 3, 8, 3, 6, 1, 3, 9, 3, 4, 3, 9, 5, 9

Offset: 0

Paul D. Hanna, Apr 12 2018

			Constant B = 0.891784622610953349715890136060239421022216970366139189336822360...
This constant equals the sum of the following infinite series.
(1) B = 1 - 1/3^2 + 1/7^3 - 1/15^4 + 1/31^5 - 1/63^6 + 1/127^7 - 1/255^8 + 1/511^9 - 1/1023^10 + 1/2047^11 - 1/4095^12 + 1/8191^13 - 1/16383^14 + ...
Also,
(2) B = 1/2 + 3/2^4 + 7^2/2^9 + 15^3/2^16 + 31^4/2^25 + 63^5/2^36 + 127^6/2^49 + 255^7/2^64 + 511^8/2^81 + 1023^9/2^100 + 2047^10/2^121 + 4095^11/2^144 + ...
Expressed in terms of powers of 1/2, we have
(3) B = 1/2 + 1/2^2 + 1/2^3 + 0/2^4 + 1/2^5 - 1/2^6 + 1/2^7 - 2/2^8 + 2/2^9 - 3/2^10 + 1/2^11 - 1/2^12 + 1/2^13 - 5/2^14 + 7/2^15 - 7/2^16 + 1/2^17 + 3/2^18 + 1/2^19 - 12/2^20 + 16/2^21 - 9/2^22 + ... + A303506(n)/2^n + ...
DECIMAL EXPANSION TO 1000 DIGITS:
B = 0.89178462261095334971589013606023942102221697036613\
91893368223601276122378175444558396786463861763710\
57439093836139343959699895448987622772561974889829\
69662500641670749267412176492387283639777757763274\
25544373227852142261116843917982062828561973242641\
82725879555976060428390970218640637206146898948643\
76158809108390913335032108295905030664382411547224\
65652844918843557563559576104945928523599994449875\
54216008705234822642417410437080548464100874227218\
61650525099561200582641085028403673931750929494032\
47382019920912650558684222318629979407415580585052\
58521100916256823999312185479604796455256751507361\
67292078514305809228767193192555896703488660216859\
38438297427435171546623099960570301622830302948131\
42393878925766586388132889946469804516455360827301\
15060737460971066848430279446396669771028830058957\
09040428237475226018628287375514768624454713520927\
57806744194504585813229218682951533161650254564160\
40305474360667599580582080941206432281172119508572\
24718465451691587123672187602470833897922105839762...

Paul D. Hanna, Table of n, a(n) for n = 0..1030

Cf. A303340 (binary), A300279.

digits = 120; B = NSum[(-1)^(n-1)/(2^n-1)^n, {n, 1, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> digits+5]; RealDigits[B, 10, digits][[1]] (* Jean-François Alcover, Apr 25 2018 *)

suminf(n=1, (-1)^(n-1)/(2^n-1)^n) \\ Michel Marcus, Apr 25 2018

This constant may be defined by the following expressions.
(1) B = Sum_{n>=1} (-1)^(n-1) / (2^n - 1)^n.
(2) B = Sum_{n>=1} (2^n - 1)^(n-1) / 2^(n^2).
(3) B = Sum_{n>=1} A303506(n)/2^n where A303506(n) = Sum_{d|n} binomial(n/d-1, d-1) * (-1)^(d-1) for n>=1.

A300280 Triangle defined by T(n,k) = Sum_{j>=0} C(j+k, k) * C((j+k)*k, n-k) / 2^(j+k+1), for n>=0, k = 0..n, as read by rows.

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 5, 10, 1, 0, 7, 57, 21, 1, 0, 9, 252, 246, 36, 1, 0, 11, 969, 2158, 710, 55, 1, 0, 13, 3414, 15927, 10260, 1635, 78, 1, 0, 15, 11329, 104883, 122125, 35085, 3255, 105, 1, 0, 17, 35992, 637252, 1273192, 611130, 96992, 5852, 136, 1, 0, 19, 110625, 3647268, 12057412, 9199386, 2321004, 230972, 9756, 171, 1, 0, 21, 331298, 19935477, 106181320, 124315310, 47518716, 7261394, 492408, 15345, 210, 1, 0, 23, 971609, 105054633, 883422885, 1546241270, 865414802, 193797618, 19669302, 963795, 23045, 253, 1

Offset: 0

Paul D. Hanna, Mar 01 2018

Is there a closed-form expression for the terms T(n,k) of this triangle?

Row sums form A300279, with g.f.: Sum_{n>=0} (1 + x*(1+x)^n)^n / 2^(n+1).

			This triangle begins:
1;
0, 1;
0, 3, 1;
0, 5, 10, 1;
0, 7, 57, 21, 1;
0, 9, 252, 246, 36, 1;
0, 11, 969, 2158, 710, 55, 1;
0, 13, 3414, 15927, 10260, 1635, 78, 1;
0, 15, 11329, 104883, 122125, 35085, 3255, 105, 1;
0, 17, 35992, 637252, 1273192, 611130, 96992, 5852, 136, 1;
0, 19, 110625, 3647268, 12057412, 9199386, 2321004, 230972, 9756, 171, 1;
0, 21, 331298, 19935477, 106181320, 124315310, 47518716, 7261394, 492408, 15345, 210, 1;
0, 23, 971609, 105054633, 883422885, 1546241270, 865414802, 193797618, 19669302, 963795, 23045, 253, 1; ...
GENERATING FUNCTIONS.
G.f.: A(x,y) = Sum_{n>=0} x^n*y^n * (1+x)^(n^2) / (2 - (1+x)^n)^(n+1).
Expanding,
G.f.: A(x,y) = 1 + x*y*(1+x)/(2 - (1+x))^2 + x^2*y^2*(1+x)^4/(2 - (1+x)^2)^3 + x^3*y^3*(1+x)^9/(2 - (1+x)^3)^4 + x^4*y^4*(1+x)^16/(2 - (1+x)^4)^5 + x^5*y^5*(1+x)^25/(2 - (1+x)^5)^6 + x^6*y^6*(1+x)^36/(2 - (1+x)^6)^7 + ...
Also, due to a series identity:
A(x,y) = 1/2 + (1 + x*y*(1+x))/2^2 + (1 + x*y*(1+x)^2)^2/2^3 + (1 + x*y*(1+x)^3)^3/2^4 + (1 + x*y*(1+x)^4)^4/2^5 + (1 + x*y*(1+x)^5)^5/2^6 + (1 + x*y*(1+x)^6)^6/2^7 + ... + (1 + x*y * (1+x)^n)^n / 2^(n+1) + ...
Explicitly,
G.f.: A(x,y) = 1 + y*x + (y^2 + 3*y)*x^2 + (y^3 + 10*y^2 + 5*y)*x^3 + (y^4 + 21*y^3 + 57*y^2 + 7*y)*x^4 + (y^5 + 36*y^4 + 246*y^3 + 252*y^2 + 9*y)*x^5 + (y^6 + 55*y^5 + 710*y^4 + 2158*y^3 + 969*y^2 + 11*y)*x^6 + (y^7 + 78*y^6 + 1635*y^5 + 10260*y^4 + 15927*y^3 + 3414*y^2 + 13*y)*x^7 + (y^8 + 105*y^7 + 3255*y^6 + 35085*y^5 + 122125*y^4 + 104883*y^3 + 11329*y^2 + 15*y)*x^8 + ...
The row sums begin
A300279 = [1, 1, 4, 16, 86, 544, 3904, 31328, 276798, 2660564, ...],
and has g.f.: Sum_{n>=0} (1 + x*(1+x)^n)^n / 2^(n+1).
RELATED TRIANGLE.
The coefficients in 1/A(x,y) forms the triangle:
1;
0, -1;
0, -3, 0;
0, -5, -4, 0;
0, -7, -38, -4, 0;
0, -9, -208, -104, -4, 0;
0, -11, -884, -1336, -202, -4, 0;
0, -13, -3268, -12112, -4768, -332, -4, 0;
0, -15, -11098, -89540, -75532, -12520, -494, -4, 0; ...

Paul D. Hanna, Table of n, a(n) for n = 0..1080, as a flattened triangle read by Rows 0..45

Cf. A300279 (row sums).

/* Must set N to a large value for accuracy: */ N=10000;
{T(n,k) = round( sum(j=0,N, binomial(j+k,k) * binomial((j+k)*k,n-k) / 2^(j+k+1)*1. ) )}
for(n=0,12, for(k=0,n, print1(T(n,k),", "));print(""))

/* Faster, without precision errors: */
{T(n,k) = my(A = sum(m=0, n, x^m * y^m * (1+x + x*O(x^n))^(m^2) / (2 - (1+x + x*O(x^n))^m )^(m+1) )); polcoeff(polcoeff(A, n,x), k,y)}
for(n=0,12, for(k=0,n, print1(T(n,k),", "));print(""))

T(n,k) = Sum_{j>=0} C(j+k, k) * C((j+k)*k, n-k) / 2^(j+k+1).
G.f. A(x,y) = Sum_{n>=0} Sum_{k=0..n} T(n,k) * x^n * y^k is given by:
(1) A(x,y) = Sum_{n>=0} (1 + x*y * (1+x)^n)^n / 2^(n+1).
(2) A(x,y) = Sum_{n>=0} x^n * y^n * (1+x)^(n^2) / (2 - (1+x)^n)^(n+1).

cp's OEIS Frontend

A300050 G.f. A(x) satisfies: A(x) = Sum_{n>=0} (1 + x*A(x)^n)^n / 2^(n+1).

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A302765 Decimal expansion of constant: B = Sum_{n>=1} (-1)^(n-1) / (2^n - 1)^n.

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A300280 Triangle defined by T(n,k) = Sum_{j>=0} C(j+k, k) * C((j+k)*k, n-k) / 2^(j+k+1), for n>=0, k = 0..n, as read by rows.

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