Real-rooted atlas

A131431 → D+1 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(10 - 21\,n + 9\,n^{2}\right)\,P_{n}(x) = \left(2 + 8\,x + 8\,x^{2} + 3\,n + 12\,n x + 12\,n x^{2} - 9\,n^{2} - 36\,n^{2} x - 36\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-10 - 20\,x - 24\,n - 48\,n x + 18\,n^{2} + 36\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 8 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
18816
2215628105
34018024080540
4654169368322082457
59680025603840256051210368
61331368570012160136807296121641553
7176215611088308004928044352197122816160380
822532001960067200140000179200134400512006400601425
9280453632256131712338688564480602112387072129024143362204496
103416200502202380807291201499904208320019046401071360317440317447932249
114088228748004039201436160351859260318727180800574464028723207659526963228107324
124811065610744865120026373607501824153162242250547223443200166707207501824181862415155298316585
135601352014976010067204576000148262403514368061501440790732807321600046858240191692804259840327680340122240
14645168562034761502592757556827547520743783041511178242313991682644561922203801601282211844808294498631687045121165450113
Technical details
Node id
n-78f041e86f6d0ee1 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-e620d4a356abb37c

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:   1;
  1:   8,    8;
  2:  21,   56,    28;
  3:  40,  180,   240,     80;
  4:  65,  416,   936,    832,    208;
  5:  96,  800,  2560,   3840,   2560,    512;
  6: 133, 1368,  5700,  12160,  13680,   7296,   1216;
  7: 176, 2156, 11088,  30800,  49280,  44352,  19712,   2816;
  8: 225, 3200, 19600,  67200, 140000, 179200, 134400,  51200,   6400;
  9: 280, 4536, 32256, 131712, 338688, 564480, 602112, 387072, 129024, 14336;
%F Formula
The row polynomials satisfy (9*n^2-21*n+10)*P(n,x) = -(9*n^2-3*n-2+(36*n^2-12*n-8)*x+(36*n^2-12*n-8)*x^2)*P(n-2,x) + (18*n^2-24*n-10+(36*n^2-48*n-20)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 8 + 8*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 8 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(9*n^2 - 3*n - 2 + (36*n^2 - 12*n - 8)*x + (36*n^2 - 12*n - 8)*x^2)*P[n - 2, x] + (18*n^2 - 24*n - 10 + (36*n^2 - 48*n - 20)*x)*P[n - 1, x])/(9*n^2 - 21*n + 10)]];
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] = 8 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (-(9*n^2 - 3*n - 2 + (36*n^2 - 12*n - 8)*x + (36*n^2 - 12*n - 8)*x^2)*v[i - 2] + (18*n^2 - 24*n - 10 + (36*n^2 - 48*n - 20)*x)*v[i - 1])/(9*n^2 - 21*n + 10)); 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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  8. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.