Real-rooted atlas

A125092 → basis A013609

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(8 + 24\,x + 24\,x^{2} + 8\,x^{3}\right)\,P_{n-3}(x) + \left(-12 - 24\,x - 12\,x^{2}\right)\,P_{n-2}(x) + \left(6 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 5 + 8\,x\)
\(P_{2}(x) = 18 + 52\,x + 36\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
15813
2185236106
356228300128712
4160832158413124004288
54322720672081604880115224064
611208256249603968035040163203136128512
728162374484672165760192640133056506248192661504
8691265536268800623616896000817152462336148480207363309568
916640175104811008217190437094404193280313958415022084170245120016187392
103942445568023500807127040140851201896652817633280111820804631040113152012390477725696
119216011601926589440223027205001216078059520865751046825984037509120136857602984960294912367525888
12212992289996817977344671334401682841602984509443841474563617095682473574401198489603906355276922886922241715470336
1348742471352324792320019563315254179840010753966081574436864172204032014075043848493056003678371841083064321943552016056327918845952
141105920173178881252392965546311681681039360369008640060517416967534804992715876761651661209602788065280109137100829297049648283648368640036205232128
Technical details
Node id
n-20024533c5d08fb3 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-352a718296b74ee0

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:     5,      8;
  2:    18,     52,     36;
  3:    56,    228,    300,     128;
  4:   160,    832,   1584,    1312,     400;
  5:   432,   2720,   6720,    8160,    4880,    1152;
  6:  1120,   8256,  24960,   39680,   35040,   16320,    3136;
  7:  2816,  23744,  84672,  165760,  192640,  133056,   50624,    8192;
  8:  6912,  65536, 268800,  623616,  896000,  817152,  462336,  148480,  20736;
  9: 16640, 175104, 811008, 2171904, 3709440, 4193280, 3139584, 1502208, 417024, 51200;
%F Formula
The row polynomials satisfy P(n,x) = (8+24*x+24*x^2+8*x^3)*P(n-3,x) - (12+24*x+12*x^2)*P(n-2,x) + (6+6*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 5 + 8*x, P(2,x) = 18 + 52*x + 36*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 5 + 8*x;
P[2, x_] := 18 + 52*x + 36*x^2;
P[n_, x_] := P[n, x] = Expand[(8 + 24*x + 24*x^2 + 8*x^3)*P[n - 3, x] - (12 + 24*x + 12*x^2)*P[n - 2, x] + (6 + 6*x)*P[n - 1, x]];
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] = 5 + 8*x; v[3] = 18 + 52*x + 36*x^2; for(n = 3, N, my(i = n + 1); v[i] = (8 + 24*x + 24*x^2 + 8*x^3)*v[i - 3] - (12 + 24*x + 12*x^2)*v[i - 2] + (6 + 6*x)*v[i - 1]); 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

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.