Real-rooted atlas

A059364 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-n - 6\,n x - 9\,n x^{2} + 4\,n^{2} + 18\,n^{2} x + 18\,n^{2} x^{2} - 4\,n^{3} - 12\,n^{3} x - 9\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-1 - 2\,n - 6\,n x + 4\,n^{2} + 6\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 6\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1369
2156654135
310573212696482754
494591082459726136972070506
5103951285384754977794365969701749602165796
613513520452149651987215445962529792015169032367416077518044
7202702536353880209204046593424576945014445865043064426753684881798403166000560
834459425715313160486520846216782747264338679828454192214810431476018420132043011842380855680145249034544
9654729075154517589901214091437584940197026481209909394215189384345815419185920152561222141719456446316906384714256704007393764498336
101374931057536377068329032449211734681523077950592843933836127815833274508129741065749533557669144527377449650619938065704163771692912002357047123200413488889224416
113162341432259274180213620926398358126194930844602689361638747885559275365631127935987656593307834821176122639537659440455916436438935622327392894400806487033291041608485369643520025197804875867904
127905853580625254623363652700281680162685095516773823625729112631574978510839951622016906401575802950483433201677173859070263904645283619460103849412202381850568115762401046340590894311682759769133569792033092941609728001661839824454306560
1321345804667687574914873979116509094911069865873559936692249093568425229763437335620757335216546714985626153464821937716982492357021587195573192826704213063275044106221153236505933446801304325853263126223251968412709596426241255353401279502720138990354760857600117926892208309814784
14619028335362937523518361046029535031094879163295372652247479095136202399610463307739904742358533987928757890611173480447359440841354489514201674055877577418641888114878784884612672188754099548670905400013999824501037191289447483267878074801965442728615264657693182726081132028439930112062545659642385920008958871542632830511616
Technical details
Node id
n-f6ae2fe52b1a553f · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-98cf788dbf2383de

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:         3,           6;
  2:        15,          66,           54;
  3:       105,         732,         1269,          648;
  4:       945,        9108,        24597,        26136,          9720;
  5:     10395,      128538,       475497,       779436,        596970,        174960;
  6:    135135,     2045214,      9651987,     21544596,      25297920,      15169032,       3674160;
  7:   2027025,    36353880,    209204046,    593424576,     945014445,     865043064,     426753684,      88179840;
  8:  34459425,   715313160,   4865208462,  16782747264,   33867982845,   41922148104,   31476018420,   13204301184,   2380855680;
  9: 654729075, 15451758990, 121409143758, 494019702648, 1209909394215, 1893843458154, 1918592015256, 1222141719456, 446316906384, 71425670400;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(4*n^3-4*n^2+n+(12*n^3-18*n^2+6*n)*x+(9*n^3-18*n^2+9*n)*x^2)*P(n-2,x) + (4*n^2-2*n-1+(6*n^2-6*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 6*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 - 4*n^2 + n + (12*n^3 - 18*n^2 + 6*n)*x + (9*n^3 - 18*n^2 + 9*n)*x^2)*P[n - 2, x] + (4*n^2 - 2*n - 1 + (6*n^2 - 6*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] = 3 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^3 - 4*n^2 + n + (12*n^3 - 18*n^2 + 6*n)*x + (9*n^3 - 18*n^2 + 9*n)*x^2)*v[i - 2] + (4*n^2 - 2*n - 1 + (6*n^2 - 6*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.