cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A301305 G.f. L(x,y) satisfies: L(x,y) = x * (1 + y*x*L'(x,y)) / (1 + x*L'(x,y)) where L'(x,y) = d/dx L(x,y), as a triangle read by rows.

Original entry on oeis.org

1, -1, 1, 3, -5, 2, -14, 33, -25, 6, 85, -261, 292, -140, 24, -621, 2363, -3516, 2546, -892, 120, 5236, -23872, 44537, -43405, 23228, -6444, 720, -49680, 264860, -596396, 733983, -532095, 226644, -52356, 5040, 521721, -3193029, 8448004, -12605668, 11586756, -6707208, 2383248, -474144, 40320, -5994155, 41506739, -126480376, 222424796, -248535142, 182793154, -88379152, 27046632, -4745376, 362880, 74701055, -578419961, 1998774636, -4056699966, 5351696394, -4791391134, 2945757656, -1226765624, 330797184, -52079040, 3628800
Offset: 1

Views

Author

Paul D. Hanna, Mar 20 2018

Keywords

Comments

Main diagonal equals the factorials.
Column 0 forms A088716, signed.
Rows sums are zeros after the initial row.
Absolute row sums equal A301388.

Examples

			G.f. L(x,y) = Sum_{n>=1, k=0..n-1} T(n,k) * x^n*y^k begins:
L(x,y) = x + (-1 + y)*x^2 + (3 - 5*y + 2*y^2)*x^3 + (-14 + 33*y - 25*y^2 + 6*y^3)*x^4 + (85 - 261*y + 292*y^2 - 140*y^3 + 24*y^4)*x^5 + (-621 + 2363*y - 3516*y^2 + 2546*y^3 - 892*y^4 + 120*y^5)*x^6 + (5236 - 23872*y + 44537*y^2 - 43405*y^3 + 23228*y^4 - 6444*y^5 + 720*y^6)*x^7 + (-49680 + 264860*y - 596396*y^2 + 733983*y^3 - 532095*y^4 + 226644*y^5 - 52356*y^6 + 5040*y^7)*x^8 + ...
where L = L(x,y) satisfies:
L = x*(1 + y*x*L') / (1 + x*L').
TRIANGLE.
This triangle of coefficients T(n,k) in L(x,y) begins:
1;
-1, 1;
3, -5, 2;
-14, 33, -25, 6;
85, -261, 292, -140, 24;
-621, 2363, -3516, 2546, -892, 120;
5236, -23872, 44537, -43405, 23228, -6444, 720;
-49680, 264860, -596396, 733983, -532095, 226644, -52356, 5040;
521721, -3193029, 8448004, -12605668, 11586756, -6707208, 2383248, -474144, 40320;
-5994155, 41506739, -126480376, 222424796, -248535142, 182793154, -88379152, 27046632, -4745376, 362880; ...
LIMITS.
In this triangle, the largest real root of the n-th row polynomial in y converges to the constant t = 2.845344903202547217277843362090557097661... (A301389).
RELATED SERIES.
exp(L(x,y)) = 1 + x + (-1 + 2*y)*x^2/2! + (13 - 24*y + 12*y^2)*x^3/3!  +  (-263 + 660*y - 540*y^2 + 144*y^3)*x^4/4!  +  (8381 - 26800*y + 31380*y^2 - 15840*y^3 + 2880*y^4)*x^5/5!  +  (-379409 + 1485870*y - 2280180*y^2 + 1706520*y^3 - 619200*y^4 + 86400*y^5)*x^6/6!  +  (22915369 - 106759128*y + 203726880*y^2 - 203269920*y^3 + 111449520*y^4 - 31691520*y^5 + 3628800*y^6)*x^7/7! + ...
satisfies: [x^n] exp(-n*L(x,y)) = ((y-1)*(n-1) - 1) * [x^(n-1)] exp(-n*L(x,y)) for n>=1.

Links

Crossrefs

Cf. A088716, A301388, A301389.

Programs

  • PARI
    {T(n,k) = my(L=x); for(i=0,n, L = x*(1 + y*x*L')/(1 + x*L' +x*O(x^n)) ); polcoeff(polcoeff(L,n,x),k,y)}
for(n=1,12, for(k=0,n-1, print1(T(n,k),", "));print(""))

Formula

G.f. L(x,y) satisfies: [x^n] exp( -n * L(x,y) ) = ((y-1)*(n-1) - 1) * [x^(n-1)] exp( -n * L(x,y) ) for n>=1.

A301385 O.g.f. A(x) satisfies: A(x) = x * (1 + 3*x*A'(x)) / (1 + x*A'(x)).

Original entry on oeis.org

1, 2, 6, 22, 94, 474, 2974, 24630, 271710, 3799570, 63378806, 1208997078, 25736584670, 602485683530, 15356903176110, 423032451327510, 12518043710674878, 395909541133928226, 13325077980379707238, 475466006418129789206, 17926802213221278261726, 712095926927360739006522, 29722097317161256669118142, 1300445348644716445771904502
Offset: 1

Views

Author

Paul D. Hanna, Mar 20 2018

Keywords

Comments

If G(x) = x*(1 + r*x*G'(x)) / (1 + x*G'(x)), then G(x) has negative coefficients if r < t, and consists entirely of nonnegative coefficients if r > t, where t = 2.8453449032025472172778433620905570976610361149... (A301389).
O.g.f. equals the logarithm of the e.g.f. of A301386.
The e.g.f. G(x) of A301386 satisfies: [x^n] G(x)^(-n) = (2*n - 3) * [x^(n-1)] G(x)^(-n) for n>=1.

Examples

			G.f.: A(x) = x + 2*x^2 + 6*x^3 + 22*x^4 + 94*x^5 + 474*x^6 + 2974*x^7 + 24630*x^8 + 271710*x^9 + 3799570*x^10 + ...
where
A(x) = x*(1 + 3*x*A'(x)) / (1 + x*A'(x)).
RELATED SERIES.
A'(x) = 1 + 4*x + 18*x^2 + 88*x^3 + 470*x^4 + 2844*x^5 + 20818*x^6 + 197040*x^7 + 2445390*x^8 + 37995700*x^9 + ...
exp(A(x)) = 1 + x + 5*x^2/2! + 49*x^3/3! + 745*x^4/4! + 16001*x^5/5! + 472621*x^6/6! + 19659025*x^7/7! + 1211940689*x^8/8! + ... + A301386*x^n/n! + ...

Links

Crossrefs

Cf. A301386, A301388, A301389, A300736, A300987.

Programs

  • Mathematica
    Rest[CoefficientList[AsymptoticDSolveValue[{A[x] == x*(1 + 3*x*A'[x])/(1 + x*A'[x]), A[1] == 1}, A[x], {x, 0, 20}], x]]  (* Requires Mathematica version 11.3 or later *) (* Vaclav Kotesovec, Mar 20 2018 *)
  • PARI
    {a(n) = my(A=x); for(i=0,n, A = x*(1 + 3*x*A')/(1 +x*A' +x*O(x^n)) ); polcoeff(A,n)}
for(n=1,30,print1(a(n),", "))

Formula

O.g.f. A(x) satisfies: [x^n] exp( -n * A(x) ) = (2*n - 3) * [x^(n-1)] exp( -n * A(x) ) for n>=1.
From Vaclav Kotesovec, Mar 20 2018: (Start)
a(n) ~ c * 2^n * n! / n^2, where c = 0.0618315205229178422646235585879521967924163...
a(n) ~ c * 2^n * n^(n - 3/2) / exp(n), where c = 0.15498863760617284891466946263730170095444214... (End)
Showing 1-2 of 2 results.

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