Real-rooted atlas

A124932 → reverse → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 + 5\,x + 10\,x^{2} + 10\,x^{3} + 5\,x^{4} + x^{5}\right)\,P_{n-5}(x) + \left(-5 - 20\,x - 30\,x^{2} - 20\,x^{3} - 5\,x^{4}\right)\,P_{n-4}(x) + \left(10 + 30\,x + 30\,x^{2} + 10\,x^{3}\right)\,P_{n-3}(x) + \left(-10 - 20\,x - 10\,x^{2}\right)\,P_{n-2}(x) + \left(5 + 5\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 8\,x\)
\(P_{2}(x) = 6 + 36\,x + 27\,x^{2}\)
\(P_{3}(x) = 10 + 96\,x + 162\,x^{2} + 64\,x^{3}\)
\(P_{4}(x) = 15 + 200\,x + 540\,x^{2} + 480\,x^{3} + 125\,x^{4}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
13811
26362769
3109616264332
4152005404801251360
5213601350192011252164992
628588283556005250226834316912
7368965292134401750012096411651253888
84512969072282244725045360246966912729163584
9551800145805376011025013608010290046080109351000477440
106624202227595040231000349272339570211200801901650013311348864
11783168326701584004455007983369507967603204009501320002395817283707904
129140564633225168080437516679522354352230630415637057150002076363369621979957376
13105509663882384384137637532432405297292615014451081033003000121121031449646137274426206208
1412063008599556784022522505945940110360251482624014594580105105005450445196560046137061740337567768320
Technical details
Node id
n-846aecace8de9c1d · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-35caaae7884ed898

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:  3,    8;
  2:  6,   36,    27;
  3: 10,   96,   162,    64;
  4: 15,  200,   540,   480,    125;
  5: 21,  360,  1350,  1920,   1125,    216;
  6: 28,  588,  2835,  5600,   5250,   2268,    343;
  7: 36,  896,  5292, 13440,  17500,  12096,   4116,   512;
  8: 45, 1296,  9072, 28224,  47250,  45360,  24696,  6912,   729;
  9: 55, 1800, 14580, 53760, 110250, 136080, 102900, 46080, 10935, 1000;
%F Formula
The row polynomials satisfy P(n,x) = (1+5*x+10*x^2+10*x^3+5*x^4+x^5)*P(n-5,x) - (5+20*x+30*x^2+20*x^3+5*x^4)*P(n-4,x) + (10+30*x+30*x^2+10*x^3)*P(n-3,x) - (10+20*x+10*x^2)*P(n-2,x) + (5+5*x)*P(n-1,x) for n >= 5, with P(0,x) = 1, P(1,x) = 3 + 8*x, P(2,x) = 6 + 36*x + 27*x^2, P(3,x) = 10 + 96*x + 162*x^2 + 64*x^3, P(4,x) = 15 + 200*x + 540*x^2 + 480*x^3 + 125*x^4.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 8*x;
P[2, x_] := 6 + 36*x + 27*x^2;
P[3, x_] := 10 + 96*x + 162*x^2 + 64*x^3;
P[4, x_] := 15 + 200*x + 540*x^2 + 480*x^3 + 125*x^4;
P[n_, x_] := P[n, x] = Expand[(1 + 5*x + 10*x^2 + 10*x^3 + 5*x^4 + x^5)*P[n - 5, x] - (5 + 20*x + 30*x^2 + 20*x^3 + 5*x^4)*P[n - 4, x] + (10 + 30*x + 30*x^2 + 10*x^3)*P[n - 3, x] - (10 + 20*x + 10*x^2)*P[n - 2, x] + (5 + 5*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] = 3 + 8*x; v[3] = 6 + 36*x + 27*x^2; v[4] = 10 + 96*x + 162*x^2 + 64*x^3; v[5] = 15 + 200*x + 540*x^2 + 480*x^3 + 125*x^4; for(n = 5, N, my(i = n + 1); v[i] = (1 + 5*x + 10*x^2 + 10*x^3 + 5*x^4 + x^5)*v[i - 5] - (5 + 20*x + 30*x^2 + 20*x^3 + 5*x^4)*v[i - 4] + (10 + 30*x + 30*x^2 + 10*x^3)*v[i - 3] - (10 + 20*x + 10*x^2)*v[i - 2] + (5 + 5*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.