Real-rooted atlas

A059364 → basis A013609 → D+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-n - 4\,n x - 4\,n x^{2} + 4\,n^{2} + 12\,n^{2} x + 8\,n^{2} x^{2} - 4\,n^{3} - 8\,n^{3} x - 4\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-1 - 2\,n - 4\,n x + 4\,n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 5 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1527
23738883
3349620332481349
43981103249452347238427613
553241183734243652152072427843840679323
681687335415866217612559359226640646094084608019489215
71414498574104344162865320193292992132263184510990089868032645120638282985
82728971451679980968445027101665951440005938504464328866777610709757441791774721032192023485940505
958053154054112366596212776869596022778961064025562807572818593960438486650256384244207733763607105536185794560958918897935
101350062050051082715019902386534673072080708367836001087995959604898486099509446019693362304242466411673660286377369679759687680371589120043013171117835
1134076275477653054023364531612329874462459629548740366385646598194986680050611206822368038452811470073620291874373043272036526799872160346929029121921928232960817496064002102349783545325
12927803081315259194823439882204144820305991796112229495586203682029502220757726425762326707160896234633477436538241537667470522854471453403308247042269132780425216457425216995328501320083046401961990553600111201396207455925
1327106205126474252944504711742499014666659176815737244309010783623714490463517481032296013158129366334333761399541042838687040109862252722776281663351626545375283226315435749691187275636785996636160139379110293012481407417715261440510117543936006328227798190930275
1484584820173727825999724289806628010545506286568482566818199409091555298344414317909950686621926799614745476548809682818614939396268224758754863073947912965237504582854505676826980688696192880640101375453151995985922662217927439384576451916039991066624423153660277555201428329123020800385511978266364453175
Technical details
Node id
n-cdfe185bf871aac9 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-80e4e402b5017e54

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:          1;
  1:          5,           2;
  2:         37,          38,            8;
  3:        349,         620,          332,           48;
  4:       3981,       10324,         9452,         3472,          384;
  5:      53241,      183734,       243652,       152072,        42784,         3840;
  6:     816873,     3541586,      6217612,      5593592,      2664064,       609408,       46080;
  7:   14144985,    74104344,    162865320,    193292992,    132263184,     51099008,     9868032,      645120;
  8:  272897145,  1679980968,   4450271016,   6595144000,   5938504464,   3288667776,  1070975744,   179177472,   10321920;
  9: 5805315405, 41123665962, 127768695960, 227789610640, 255628075728, 185939604384, 86650256384, 24420773376, 3607105536, 185794560;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(4*n^3-4*n^2+n+(8*n^3-12*n^2+4*n)*x+(4*n^3-8*n^2+4*n)*x^2)*P(n-2,x) + (4*n^2-2*n-1+(4*n^2-4*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 5 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 - 4*n^2 + n + (8*n^3 - 12*n^2 + 4*n)*x + (4*n^3 - 8*n^2 + 4*n)*x^2)*P[n - 2, x] + (4*n^2 - 2*n - 1 + (4*n^2 - 4*n)*x)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 5 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^3 - 4*n^2 + n + (8*n^3 - 12*n^2 + 4*n)*x + (4*n^3 - 8*n^2 + 4*n)*x^2)*v[i - 2] + (4*n^2 - 2*n - 1 + (4*n^2 - 4*n)*x)*v[i - 1])/(n - 1)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.