Real-rooted atlas

A196347 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4 - 6\,x + 18\,n x + 9\,n^{2} x^{2}\right)\,P_{n}(x) = \left(8\,n + 36\,n x + 54\,n x^{2} + 27\,n x^{3} + 36\,n^{2} x + 90\,n^{2} x^{2} + 54\,n^{2} x^{3} + 18\,n^{3} x^{2} + 27\,n^{3} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
121214
2896162266
348864291625926420
43849216466568294448600187800
538401152007776002073600243000010497606450000
64608016588801399680049766400874800007558272025719120254250000
764512027095040274337280121927680028576800003703553280252047376070543872011308500000
810321920495452160585252864031213486080914457600001580182732801613103206409029615616021427701120560070000000
9185794560100329062401354442342408427641241602962842624000639974006784087107573145607313988648960347128758144071425670400030561300000000
10371589120022295347200033861058560002407897497600098761420800000255989602713600435537865728000487599243264000347128758144000142851340800000259275183552001821487500000000
11817496064005395474022400910486241280007283889930240003414323404800000103249139761152002108003270123520029499754217472000280017198236160001728501223680000062744594419584001018244357222400117723375000000000
1219619905536001412633198592002622200374886400233084477767680001229156425728000004247964607315968001011841569659289600169918584292638720020161238273003520001659361174732800000903522159642009600293254374880051200430208240926464008200095750000000000
1351011754393600397891684270080080573066064691200787825534854758400461616524328960000017947650465909964800488574929349771264009572080248485314560013628997072550379520014021601926492160000010179682998633308160049559989354728652800145410385433144832001945864966652006400612364252500000000000
141428329123020800119979646333747200263205349144657920028075237242096844800180953677536952320000781719886959634022400239401715381387919360053603649391517761536008904278087399581286400109929359103698534400009976089338660641996800647583860901787729920028500435544896387072007627790669275865088009381848946357888000048803667412500000000000
Technical details
Node id
n-82e736284320e8de · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-56805ed74e931bf6

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:         2,          12;
  2:         8,          96,          162;
  3:        48,         864,         2916,         2592;
  4:       384,        9216,        46656,        82944,         48600;
  5:      3840,      115200,       777600,      2073600,       2430000,       1049760;
  6:     46080,     1658880,     13996800,     49766400,      87480000,      75582720,      25719120;
  7:    645120,    27095040,    274337280,   1219276800,    2857680000,    3703553280,    2520473760,     705438720;
  8:  10321920,   495452160,   5852528640,  31213486080,   91445760000,  158018273280,  161310320640,   90296156160,   21427701120;
  9: 185794560, 10032906240, 135444234240, 842764124160, 2962842624000, 6399740067840, 8710757314560, 7313988648960, 3471287581440, 714256704000;
%F Formula
The row polynomials satisfy (4+(18*n-6)*x+9*n^2*x^2)*P(n,x) = (8*n+(36*n^2+36*n)*x+(18*n^3+90*n^2+54*n)*x^2+(27*n^3+54*n^2+27*n)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n + (36*n^2 + 36*n)*x + (18*n^3 + 90*n^2 + 54*n)*x^2 + (27*n^3 + 54*n^2 + 27*n)*x^3)*P[n - 1, x])/(4 + (18*n - 6)*x + 9*n^2*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((8*n + (36*n^2 + 36*n)*x + (18*n^3 + 90*n^2 + 54*n)*x^2 + (27*n^3 + 54*n^2 + 27*n)*x^3)*v[i - 1])/(4 + (18*n - 6)*x + 9*n^2*x^2)); 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

Same fingerprint ?

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.