Real-rooted atlas

A038281 → basis A013611 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(23 + 12\,x + 12\,n\right)\,P_{n}(x) = \left(385 + 552\,x + 144\,x^{2} + 132\,n + 144\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 385 + 552\,x + 144\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
03855521441081
15687138609936172831211
27852927297633264015897620736863857
31039511471174081892806652800238464024883223226803
41336723374844792169622640196542720119750400343388162985984611452585
5168298295112284757231434812644749433920412739712020118067204807434243583180815839840123
62085127297161566363205389668345610059140044811398641408079246024704321889075206593052672429981696405174228193
725508706839225193748076872458361280194028060441627159678120962462106544128142642844467249662885888089006211072515978035210258739071811

All 15 rows

Technical details
Node id
n-aee3f5e748e8b8e4 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-29d09fbfedb73813

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:           385,            552,             144;
  1:          5687,          13860,            9936,            1728;
  2:         78529,         272976,          332640,          158976,            20736;
  3:       1039511,        4711740,         8189280,         6652800,          2384640,           248832;
  4:      13367233,       74844792,       169622640,       196542720,        119750400,         34338816,          2985984;
  5:     168298295,     1122847572,      3143481264,      4749433920,       4127397120,       2011806720,        480743424,        35831808;
  6:    2085127297,    16156636320,     53896683456,    100591400448,     113986414080,      79246024704,      32188907520,      6593052672,       429981696;
  7:   25508706839,   225193748076,    872458361280,   1940280604416,    2715967812096,    2462106544128,    1426428444672,    496628858880,     89006211072,      5159780352;
  8:  308891147521,  3061044820680,  13511624884560,  34898334451200,   58208418132480,   65183227490304,   49242130882560,  24453059051520,   7449432883200,   1186749480960,    61917364224;
  9: 3709051717943, 40773631472772, 202028958164880, 594511494920640, 1151645036889600, 1536702238697472, 1434031004786688, 928565896642560, 403475474350080, 109258348953600, 15665093148672, 743008370688;
%F Formula
The row polynomials satisfy (12*n+23+12*x)*P(n,x) = (132*n+385+(144*n+552)*x+144*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 385 + 552*x + 144*x^2.
%t Mathematica
P[0, x_] := 385 + 552*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((132*n + 385 + (144*n + 552)*x + 144*x^2)*P[n - 1, x])/(12*n + 23 + 12*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 385 + 552*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((132*n + 385 + (144*n + 552)*x + 144*x^2)*v[i - 1])/(12*n + 23 + 12*x)); 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.