Real-rooted atlas

A013609 → D → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-1 + 8\,x - 4\,x^{2} + n - 14\,n x + 12\,n x^{2} + 6\,n^{2} x - 12\,n^{2} x^{2} + 4\,n^{3} x^{2}\right)\,P_{n}(x) = \left(n - 4\,n x^{2} + 6\,n^{2} x + 12\,n^{2} x^{2} + 4\,n^{3} x^{2} + 8\,n^{3} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
122
243236
3696216318
4819286410242088
51032021605120400011610
612480432015360240001382457996
714672756035840840009676843904268758
81689612096716802240003870723512321310721178064
9181152181441290245040001161216158054411796483732484946994
102014402592021504010080002903040526848058982403732480102400020076660
11221760356403379201848000638668814488320216268802052864011264000272588879243758
122421124752050688031680001277337634771968648806408211456067584000327106567077888305637624
13262496617767321605148000237219847533926416868966426687232029286400021261926492012544179978241156061322
142829127862410250248008000415134721506785283936092167472424961025024000992223232644087808251969536449576964300420572
Technical details
Node id
n-948826d8d47a4c2c · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-181bf1738e48841f

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:  0;
  1:  2;
  2:  4,   32;
  3:  6,   96,   216;
  4:  8,  192,   864,   1024;
  5: 10,  320,  2160,   5120,   4000;
  6: 12,  480,  4320,  15360,  24000,   13824;
  7: 14,  672,  7560,  35840,  84000,   96768,   43904;
  8: 16,  896, 12096,  71680, 224000,  387072,  351232,  131072;
  9: 18, 1152, 18144, 129024, 504000, 1161216, 1580544, 1179648, 373248;
%F Formula
The row polynomials satisfy (n-1+(6*n^2-14*n+8)*x+(4*n^3-12*n^2+12*n-4)*x^2)*P(n,x) = (n+6*n^2*x+(4*n^3+12*n^2-4*n)*x^2+8*n^3*x^3)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((n + 6*n^2*x + (4*n^3 + 12*n^2 - 4*n)*x^2 + 8*n^3*x^3)*P[n - 1, x])/(n - 1 + (6*n^2 - 14*n + 8)*x + (4*n^3 - 12*n^2 + 12*n - 4)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((n + 6*n^2*x + (4*n^3 + 12*n^2 - 4*n)*x^2 + 8*n^3*x^3)*v[i - 1])/(n - 1 + (6*n^2 - 14*n + 8)*x + (4*n^3 - 12*n^2 + 12*n - 4)*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.