A190726 -id:A190726 - cp's OEIS frontend

A200536 Triangle, read by rows of 2n+1 terms, where row n lists the coefficients in (1+3x+2*x^2)^n.

1, 1, 3, 2, 1, 6, 13, 12, 4, 1, 9, 33, 63, 66, 36, 8, 1, 12, 62, 180, 321, 360, 248, 96, 16, 1, 15, 100, 390, 985, 1683, 1970, 1560, 800, 240, 32, 1, 18, 147, 720, 2355, 5418, 8989, 10836, 9420, 5760, 2352, 576, 64, 1, 21, 203, 1197, 4809, 13923, 29953, 48639, 59906, 55692, 38472, 19152, 6496, 1344, 128

Offset: 0

Paul D. Hanna, Nov 18 2011

			The triangle begins:
1;
1, 3, 2;
1, 6, 13, 12, 4;
1, 9, 33, 63, 66, 36, 8;
1, 12, 62, 180, 321, 360, 248, 96, 16;
1, 15, 100, 390, 985, 1683, 1970, 1560, 800, 240, 32;
1, 18, 147, 720, 2355, 5418, 8989, 10836, 9420, 5760, 2352, 576, 64;
1, 21, 203, 1197, 4809, 13923, 29953, 48639, 59906, 55692, 38472, 19152, 6496, 1344, 128;
1, 24, 268, 1848, 8806, 30744, 81340, 166344, 265729, 332688, 325360, 245952, 140896, 59136, 17152, 3072, 256; ...
where row n equals the coefficients in (1+x)^n*(1+2*x)^n.

Paul D. Hanna, Table of n, a(n) for n = 0..1295; Rows n = 0..35, flattened.
We-jin Woan, The Lagrange Inversion formula and divisibility properties, JIS 10 (2007) 07.7.8, Example 3.

Cf. A001850 (central Delannoy numbers), A006318, A190726; related triangle: A118384.

Cf. A200537.

{T(n,k)=polcoeff((1+3*x+2*x^2+x*O(x^k))^n,k)}
{for(n=0,10,for(k=0,2*n,print1(T(n,k),","));print(""))}

Central terms in rows form the central Delannoy numbers: T(n,n) = A001850(n).
T(2*n,n) = A190726(n).
T(n,n+1) = n*A006318(n), where A006318 form the large Schroeder numbers.

A316911 Define K(n) = Integral_{t=0..1} (-1/2)^n/(1+t)((1-t)^2t^2/(1+t))^ndt and write K(n) = d(n)log(2) - a(n)/c(n) where a(n), d(n), c(n) are positive integers; sequence gives a(n).

Original entry on oeis.org

0, 25, 1719, 143731, 64456699, 1846991851, 781688106621, 445837607665267, 611642484654021, 674842075634295726569, 9142845536119405749427, 38984536004906714808649, 80321414381403813427242343, 342487507476162248453574514441, 562411667990487545372378396727201

Offset: 0

Bradley Klee, Jul 16 2018

As n goes to infinity, integral value K(n) goes to zero. Given a rational approximant r(n)=a(n)/c(n)/d(n)=p(n)/q(n) to irrational number log(2), the quality M(n) is defined as, M(n)=-log(|r(n)-log(2)|)/log(q(n)) (Cf. Beukers Link). For this approximation, we can easily measure M(n) over n=5,000..20,000, and estimate that M(n)~1.14... to the 99% confidence level (Cf. Histogram Link).

			{a(10),c(10),d(10)}={9142845536119405749427,307660953600,42872967012}.
r(10)=a(10)/c(10)/d(10)=9142845536119405749427/13190337914573262643200.
r(10)=0.693147180559945309417232121402...
log(2)=0.693147180559945309417232121458...
M(10)=-log(|r(10)-log(2)|)/log(13190337914573262643200)=1.27...

F. Beukers, A rational approach to Pi, Nieuw archief voor wiskunde 5/1 No. 4, December 2000, p. 378.
Bradley Klee, Quality Histogram.

Integer Part: A190726. Denominators: A316912. Similar Pi approximation: A123178, A305997, A305998.

FracData[n0_]:=RecurrenceTable[{2*(n-1)*(2*n-3)*(2*n-1)*(33*n-8)*a[n-2]+ 9*(2*n-1)*(693*n^3-1554*n^2+989*n-160)*a[n-1] -3*n*(3*n-2)*(3*n-1)*(33*n-41)*a[n] == 0, a[0]==0, a[1]==25/6}, a, {n, 0, n0}]
Numerator[FracData[5000]]

Define G(x) = Sum_{n>0} A316911(n)/A316912(n)*x^n, and G^(k)(x) = d^k/dx^k G(x). Period G(x) satisfies a nonhomogeneous differential equation: -225+112*x = Sum_{j=0..5,k=0..3} M_{j,k} x^j G^(k)(x), with integer matrix M as in A190726.

A316912 Define K(n) = Integral_{t=0..1} (-1/2)^n/(1+t)((1-t)^2t^2/(1+t))^ndt and write K(n) = d(n)log(2) - b(n)/a(n) where a(n), d(n), b(n) are positive integers; sequence gives a(n).

Original entry on oeis.org

1, 6, 40, 288, 10560, 24024, 792064, 34728960, 3627008, 302356454400, 307660953600, 98050867200, 15038824120320, 4757532010463232, 577952036826644480, 26189033224273920, 358597702262241361920, 244498433360619110400, 143982410756809031680

Offset: 0

Bradley Klee, Jul 16 2018

Integer Part: A190726. Numerators: A316911. Similar Pi Approximation: A123178, A305997, A305998.

FracData[n0_]:=RecurrenceTable[{2*(n-1)*(2*n-3)*(2*n-1)*(33*n-8)*a[n-2]+ 9*(2*n-1)*(693*n^3-1554*n^2+989*n-160)*a[n-1] -3*n*(3*n-2)*(3*n-1)*(33*n-41)*a[n] == 0, a[0]==0, a[1]==25/6}, a, {n, 0, n0}]
Denominator[FracData[5000]]

cp's OEIS Frontend

A200536 Triangle, read by rows of 2n+1 terms, where row n lists the coefficients in (1+3x+2*x^2)^n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A316911 Define K(n) = Integral_{t=0..1} (-1/2)^n/(1+t)((1-t)^2t^2/(1+t))^ndt and write K(n) = d(n)log(2) - a(n)/c(n) where a(n), d(n), c(n) are positive integers; sequence gives a(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A316912 Define K(n) = Integral_{t=0..1} (-1/2)^n/(1+t)((1-t)^2t^2/(1+t))^ndt and write K(n) = d(n)log(2) - b(n)/a(n) where a(n), d(n), b(n) are positive integers; sequence gives a(n).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

cp's OEIS Frontend

A200536 Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A316911 Define K(n) = Integral_{t=0..1} (-1/2)^n/(1+t)*((1-t)^2*t^2/(1+t))^n*dt and write K(n) = d(n)*log(2) - a(n)/c(n) where a(n), d(n), c(n) are positive integers; sequence gives a(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A316912 Define K(n) = Integral_{t=0..1} (-1/2)^n/(1+t)*((1-t)^2*t^2/(1+t))^n*dt and write K(n) = d(n)*log(2) - b(n)/a(n) where a(n), d(n), b(n) are positive integers; sequence gives a(n).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A200536 Triangle, read by rows of 2n+1 terms, where row n lists the coefficients in (1+3x+2*x^2)^n.

A316911 Define K(n) = Integral_{t=0..1} (-1/2)^n/(1+t)((1-t)^2t^2/(1+t))^ndt and write K(n) = d(n)log(2) - a(n)/c(n) where a(n), d(n), c(n) are positive integers; sequence gives a(n).

A316912 Define K(n) = Integral_{t=0..1} (-1/2)^n/(1+t)((1-t)^2t^2/(1+t))^ndt and write K(n) = d(n)log(2) - b(n)/a(n) where a(n), d(n), b(n) are positive integers; sequence gives a(n).