Real-rooted atlas

A038338 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 + 5\,x + 4\,x^{2} + 3\,n x + 4\,n x^{2} + n^{2} x^{2}\right)\,P_{n}(x) = \left(24 + 216\,x + 408\,x^{2} + 216\,x^{3} + 72\,n x + 216\,n x^{2} + 144\,n x^{3} + 24\,n^{2} x^{2} + 24\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 576 + 4608\,x + 5184\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
05764608518410368
113824165888373248221184774144
233177653084161791590421233664829440053084160
3796262415925248071663616012740198409953280002866544643439853568
41911029764586471424257989017606115295232071663616000412782428169364045824214035333120
54586471424128421199872866843099136256842399744040131624960003467372396544157315969843229353417113612915503529984
611007531417635224100536322773897917235298627481501696192631799808000221911833378816151023331049472563585608581128916100448256760840571584512
72641807540224951050714480648559456430325763550589334061056832169375170560011983239002456064108736798355619846086724572676096192587769682329626418075402240043959677469327360

All 15 rows

Technical details
Node id
n-d0334e90234e34f3 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-d42e6339eab21b9c

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:              576,              4608,               5184;
  1:            13824,            165888,             373248,              221184;
  2:           331776,           5308416,           17915904,            21233664,              8294400;
  3:          7962624,         159252480,          716636160,          1274019840,            995328000,            286654464;
  4:        191102976,        4586471424,        25798901760,         61152952320,          71663616000,          41278242816,           9364045824;
  5:       4586471424,      128421199872,       866843099136,       2568423997440,        4013162496000,        3467372396544,        1573159698432,         293534171136;
  6:     110075314176,     3522410053632,     27738979172352,      98627481501696,      192631799808000,      221911833378816,      151023331049472,       56358560858112,        8916100448256;
  7:    2641807540224,    95105071448064,    855945643032576,    3550589334061056,     8321693751705600,    11983239002456064,    10873679835561984,     6086724572676096,     1925877696823296,     264180754022400;
  8:   63403380965376,  2536135238615040,  25678369290977280,  121734491453521920,   332867750068224000,   575195472117891072,   652420790133719040,   486937965814087680,   231105323618795520,   63403380965376000,    7671809096810496;
  9: 1521681143169024, 66953970299437056, 753232165868666880, 4017238217966223360, 12553869431144448000, 25308600773187207168, 34447817719060365312, 32137905743729786880, 20337268478454005760, 8369246287429632000, 2025357601557970944, 219122084616339456;
%F Formula
The row polynomials satisfy (1+(3*n+5)*x+(n^2+4*n+4)*x^2)*P(n,x) = (24+(72*n+216)*x+(24*n^2+216*n+408)*x^2+(24*n^2+144*n+216)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 576 + 4608*x + 5184*x^2.
%t Mathematica
P[0, x_] := 576 + 4608*x + 5184*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((24 + (72*n + 216)*x + (24*n^2 + 216*n + 408)*x^2 + (24*n^2 + 144*n + 216)*x^3)*P[n - 1, x])/(1 + (3*n + 5)*x + (n^2 + 4*n + 4)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 576 + 4608*x + 5184*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((24 + (72*n + 216)*x + (24*n^2 + 216*n + 408)*x^2 + (24*n^2 + 144*n + 216)*x^3)*v[i - 1])/(1 + (3*n + 5)*x + (n^2 + 4*n + 4)*x^2)); 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. [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
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.