Real-rooted atlas

A125092 → reverse → (θ+1)²

palindromicγ-positivenonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 + 5\,x + 10\,x^{2} + 10\,x^{3} + 5\,x^{4} + x^{5}\right)\,P_{n-5}(x) + \left(-5 - 20\,x - 30\,x^{2} - 20\,x^{3} - 5\,x^{4}\right)\,P_{n-4}(x) + \left(10 + 30\,x + 30\,x^{2} + 10\,x^{3}\right)\,P_{n-3}(x) + \left(-10 - 20\,x - 10\,x^{2}\right)\,P_{n-2}(x) + \left(5 + 5\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 4\,x\)
\(P_{2}(x) = 9 + 32\,x + 9\,x^{2}\)
\(P_{3}(x) = 16 + 108\,x + 108\,x^{2} + 16\,x^{3}\)
\(P_{4}(x) = 25 + 256\,x + 486\,x^{2} + 256\,x^{3} + 25\,x^{4}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1448
2932950
31610810816248
425256486256251048
53650014401440500363952
6498643375512033758644913696
764137268041400014000680413726444480
88120481234832256437503225612348204881137216
91002916207366585611340011340065856207362916100406016
101214000328051228802572503265922572501228803280540001211160704
111445324495002138405280008149688149685280002138404950053241443223552
12169691271874352000100237518247682218524182476810023753520007187469121698734720
131968788101088553696178750037528925381376538137637528921787500553696101088878819623171072
142251097613841183865630280257207200119189071405747211918907720720030280258386561384111097622560342272
Technical details
Node id
n-b08e50a4d296f47f · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-9e8dfd039a31d5a7

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: yes
  • γ-positive (gamma_positive): yes
  • 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:   4,    4;
  2:   9,   32,     9;
  3:  16,  108,   108,    16;
  4:  25,  256,   486,   256,     25;
  5:  36,  500,  1440,  1440,    500,     36;
  6:  49,  864,  3375,  5120,   3375,    864,    49;
  7:  64, 1372,  6804, 14000,  14000,   6804,  1372,    64;
  8:  81, 2048, 12348, 32256,  43750,  32256, 12348,  2048,   81;
  9: 100, 2916, 20736, 65856, 113400, 113400, 65856, 20736, 2916, 100;
%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) = 4 + 4*x, P(2,x) = 9 + 32*x + 9*x^2, P(3,x) = 16 + 108*x + 108*x^2 + 16*x^3, P(4,x) = 25 + 256*x + 486*x^2 + 256*x^3 + 25*x^4.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 4*x;
P[2, x_] := 9 + 32*x + 9*x^2;
P[3, x_] := 16 + 108*x + 108*x^2 + 16*x^3;
P[4, x_] := 25 + 256*x + 486*x^2 + 256*x^3 + 25*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] = 4 + 4*x; v[3] = 9 + 32*x + 9*x^2; v[4] = 16 + 108*x + 108*x^2 + 16*x^3; v[5] = 25 + 256*x + 486*x^2 + 256*x^3 + 25*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.