Real-rooted atlas

A038221 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4 - 2\,x + 6\,n x + n^{2} x^{2}\right)\,P_{n}(x) = \left(24 + 36\,x + 18\,x^{2} + 3\,x^{3} + 36\,n x + 30\,n x^{2} + 6\,n x^{3} + 6\,n^{2} x^{2} + 3\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
161218
23614481261
3216129614584323402
41296103681749610368202541553
5777677760174960155520607508748485514
646656559872157464018662401093500314928357215491557
727993639191041322697619595520153090006613488150028213996860584274
8167961626873856105815808188116992183708000105815808360067686718464531441655266753
9100776961813985288162933761693052928198404640014285134086481218241813985282869781419683006973568802
10604661761209323520612220032014511882240198404640001714216089697218273603627970560860934420118098000714492973222472421
113627970567981535232448961356801197230284801870672320001885637698561283281211525986151424018940557240389723400047156531425509168760118999418
1221767823365224277606432325217689695778422784016836050880001939513061376153993745382486200580505634093003032093533616000169763513041836660096898135297813883842641
131306069401633957804441622921517998087470716977152145912440960001891025234841617159303056896112060754657285318508472992182390551200044138513390471629743744700545526231248730879645129287642
147836416409621941965946881604506259865657049111461888122566450406400176495688585216180172682097408134472905588736744591186218883064161260160092690878119842005632824832294229121004262489338721076168025805771467580293
Technical details
Node id
n-9e630cc679459c11 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-e1337d43fda52a72

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:        6,        12;
  2:       36,       144,        81;
  3:      216,      1296,      1458,        432;
  4:     1296,     10368,     17496,      10368,       2025;
  5:     7776,     77760,    174960,     155520,      60750,       8748;
  6:    46656,    559872,   1574640,    1866240,    1093500,     314928,     35721;
  7:   279936,   3919104,  13226976,   19595520,   15309000,    6613488,   1500282,    139968;
  8:  1679616,  26873856, 105815808,  188116992,  183708000,  105815808,  36006768,   6718464,   531441;
  9: 10077696, 181398528, 816293376, 1693052928, 1984046400, 1428513408, 648121824, 181398528, 28697814, 1968300;
%F Formula
The row polynomials satisfy (4+(6*n-2)*x+n^2*x^2)*P(n,x) = (24+(36*n+36)*x+(6*n^2+30*n+18)*x^2+(3*n^2+6*n+3)*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[((24 + (36*n + 36)*x + (6*n^2 + 30*n + 18)*x^2 + (3*n^2 + 6*n + 3)*x^3)*P[n - 1, x])/(4 + (6*n - 2)*x + 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] = ((24 + (36*n + 36)*x + (6*n^2 + 30*n + 18)*x^2 + (3*n^2 + 6*n + 3)*x^3)*v[i - 1])/(4 + (6*n - 2)*x + 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. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [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
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.