Real-rooted atlas

A048993 → (θ+1)² → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(9 + 18\,x + 9\,x^{2}\right)\,P_{n-2}(x) + \left(4 + 12\,x + 12\,x^{2} + 4\,x^{3}\right)\,P_{n-2}'(x) + \left(3 + 10\,x + 11\,x^{2} + 4\,x^{3}\right)\,P_{n-2}''(x) + \left(3 + 4\,x\right)\,P_{n-1}(x) + \left(-2 - 6\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1448
21322944
3471067516244
4188518501196251428
5825265431951760430368900
6393714376204841432050608344959060
720270824081346941130365162512558147764416132
8111835499154916071891492495425158172278462440813102516
96574233187102646634171274164641840182498442667856496381610024392196
1040976222139513047426199582049604330108020144250585717310388401066955710121201577780
11269658671506140943613932634874939964066666452178342547557095716797528232501518997082391441745859972
12186685725110912591228592661004196482928387072290023405573409415866462514196084386422748504503220031153216915804895540
131355314108852448563623465019552371661073683749051735025210186716115188853783588239512756082899105827150953509752353615730196149190740228
1410289242825682358520221995245594253387674741363704336426702737746931921398518601484974618563212289274712208597389023861337517814888821366209862251465256029492
Technical details
Node id
n-69a9113b6c2e01a7 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-801f1db599e58f37

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:      4,       4;
  2:     13,      22,       9;
  3:     47,     106,      75,      16;
  4:    188,     518,     501,     196,      25;
  5:    825,    2654,    3195,    1760,     430,      36;
  6:   3937,   14376,   20484,   14320,    5060,     834,     49;
  7:  20270,   82408,  134694,  113036,   51625,   12558,   1477,    64;
  8: 111835,  499154,  916071,  891492,  495425,  158172,  27846,  2440,   81;
  9: 657423, 3187102, 6466341, 7127416, 4641840, 1824984, 426678, 56496, 3816, 100;
%F Formula
The row polynomials satisfy P(n,x) = (9+18*x+9*x^2)*P(n-2,x) + (4+12*x+12*x^2+4*x^3)*d/dx P(n-2,x) + (3+10*x+11*x^2+4*x^3)*d^2/dx^2 P(n-2,x) + (3+4*x)*P(n-1,x) - (2+6*x+4*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 4*x;
P[n_, x_] := P[n, x] = Expand[(9 + 18*x + 9*x^2)*P[n - 2, x] + (4 + 12*x + 12*x^2 + 4*x^3)*D[P[n - 2, x], x] + (3 + 10*x + 11*x^2 + 4*x^3)*D[P[n - 2, x], {x, 2}] + (3 + 4*x)*P[n - 1, x] - (2 + 6*x + 4*x^2)*D[P[n - 1, x], 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; for(n = 2, N, my(i = n + 1); v[i] = (9 + 18*x + 9*x^2)*v[i - 2] + (4 + 12*x + 12*x^2 + 4*x^3)*deriv(v[i - 2], x) + (3 + 10*x + 11*x^2 + 4*x^3)*deriv(deriv(v[i - 2], x), x) + (3 + 4*x)*v[i - 1] - (2 + 6*x + 4*x^2)*deriv(v[i - 1], x)); 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. [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.