Real-rooted atlas

A038208 → Veronese 2 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-36 - 12\,x + 36\,n + 10\,n x + 24\,n^{2} x - n^{2} x^{2} - 22\,n^{3} x - n^{3} x^{2} + n^{4} x^{2} + n^{5} x^{2}\right)\,P_{n}(x) = \left(-72 - 312\,x + 8\,x^{2} - 8\,x^{3} + 72\,n + 308\,n x + 124\,n x^{2} - 16\,n x^{3} + 48\,n^{2} x - 58\,n^{2} x^{2} - 2\,n^{2} x^{3} - 44\,n^{3} x - 78\,n^{3} x^{2} + 14\,n^{3} x^{3} + 2\,n^{4} x^{2} + 10\,n^{4} x^{3} + 2\,n^{5} x^{2} + 2\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(144\,x - 176\,x^{2} + 48\,x^{3} - 16\,x^{4} - 168\,n x^{2} + 216\,n x^{3} - 48\,n x^{4} - 88\,n^{2} x^{2} + 140\,n^{2} x^{3} - 52\,n^{2} x^{4} + 24\,n^{3} x^{3} - 24\,n^{3} x^{4} + 4\,n^{4} x^{3} - 4\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
122
241620
3896104
416384144544
532128014402752
66438408640102413568
712810752403201433665536
8256286721612801146886400311296
9512737285806086881281152001458176
101024184320193536034406401152000368646750208
112048450560608256015138816844800081100830932992
12409610813441824768060555264506880009732096200704140509184
138192255590452715520224919552263577600843448325218304633339904
1416384596377614760345678721843212300288005904138247305625610485762835349504
Technical details
Node id
n-2a832b7b6a769b3d · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-1f0181c04d99414a

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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;
  2:   4,    16;
  3:   8,    96;
  4:  16,   384,    144;
  5:  32,  1280,   1440;
  6:  64,  3840,   8640,   1024;
  7: 128, 10752,  40320,  14336;
  8: 256, 28672, 161280, 114688,   6400;
  9: 512, 73728, 580608, 688128, 115200;
%F Formula
The row polynomials satisfy (36*n-36-(22*n^3-24*n^2-10*n+12)*x+(n^5+n^4-n^3-n^2)*x^2)*P(n,x) = (72*n-72-(44*n^3-48*n^2-308*n+312)*x+(2*n^5+2*n^4-78*n^3-58*n^2+124*n+8)*x^2+(2*n^5+10*n^4+14*n^3-2*n^2-16*n-8)*x^3)*P(n-1,x) + (144*x-(88*n^2+168*n+176)*x^2+(4*n^4+24*n^3+140*n^2+216*n+48)*x^3-(4*n^4+24*n^3+52*n^2+48*n+16)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((72*n - 72 - (44*n^3 - 48*n^2 - 308*n + 312)*x + (2*n^5 + 2*n^4 - 78*n^3 - 58*n^2 + 124*n + 8)*x^2 + (2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8)*x^3)*P[n - 1, x] + (144*x - (88*n^2 + 168*n + 176)*x^2 + (4*n^4 + 24*n^3 + 140*n^2 + 216*n + 48)*x^3 - (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^4)*D[P[n - 1, x], x])/(36*n - 36 - (22*n^3 - 24*n^2 - 10*n + 12)*x + (n^5 + n^4 - n^3 - 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; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((72*n - 72 - (44*n^3 - 48*n^2 - 308*n + 312)*x + (2*n^5 + 2*n^4 - 78*n^3 - 58*n^2 + 124*n + 8)*x^2 + (2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8)*x^3)*v[i - 1] + (144*x - (88*n^2 + 168*n + 176)*x^2 + (4*n^4 + 24*n^3 + 140*n^2 + 216*n + 48)*x^3 - (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^4)*deriv(v[i - 1], x))/(36*n - 36 - (22*n^3 - 24*n^2 - 10*n + 12)*x + (n^5 + n^4 - n^3 - 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.