Real-rooted atlas

A130123 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - 3\,x + 9\,n x + 9\,n^{2} x^{2}\right)\,P_{n}(x) = \left(2 + 18\,x + 54\,x^{2} + 54\,x^{3} + 18\,n x + 90\,n x^{2} + 108\,n x^{3} + 18\,n^{2} x^{2} + 54\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
122426
2496324424
38288194434565696
4167687776276483240068608
532192025920138240324000279936770048
6644608777605529601944000335923222861448224768
7128107522177281935360907200023514624320060161791590484672512
825624576580608619315236288000125411328256048128286654464136048896847249408
95125529614929921857945613063680056435097615362887682579890176244888012810077696008287944704
10102412288037324805308416043545600022574039047681443840171992678402448880128020155392000731640729679591112704
112048270336912384014598144013685760008277147648337983528969459597312017958454272022170931200016096096051252242776064752693018624
1240965898242189721638928384041057280002837879193613519341158445406067097610775072563201773674496000193153152614412538266255363678762147847026566496256
13819212779525175705610121379841186099200092231073792502146957312196759624089656030377328641152888422400016739939893248162997461319689564781584384255989602713664871186038784
141638427525121207664642576351232332107776002869411184641757514350592787038496358426147509420032645617516544001171795792527361521309638983681339069421813767167708875980817631936921600593186523185152
Technical details
Node id
n-7684de62f3947998 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-10c18c8945c6e24a

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,    24;
  2:   4,    96,     324;
  3:   8,   288,    1944,     3456;
  4:  16,   768,    7776,    27648,     32400;
  5:  32,  1920,   25920,   138240,    324000,    279936;
  6:  64,  4608,   77760,   552960,   1944000,   3359232,    2286144;
  7: 128, 10752,  217728,  1935360,   9072000,  23514624,   32006016,   17915904;
  8: 256, 24576,  580608,  6193152,  36288000, 125411328,  256048128,  286654464,  136048896;
  9: 512, 55296, 1492992, 18579456, 130636800, 564350976, 1536288768, 2579890176, 2448880128, 1007769600;
%F Formula
The row polynomials satisfy (1+(9*n-3)*x+9*n^2*x^2)*P(n,x) = (2+(18*n+18)*x+(18*n^2+90*n+54)*x^2+(54*n^2+108*n+54)*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[((2 + (18*n + 18)*x + (18*n^2 + 90*n + 54)*x^2 + (54*n^2 + 108*n + 54)*x^3)*P[n - 1, x])/(1 + (9*n - 3)*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] = ((2 + (18*n + 18)*x + (18*n^2 + 90*n + 54)*x^2 + (54*n^2 + 108*n + 54)*x^3)*v[i - 1])/(1 + (9*n - 3)*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. [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.