Real-rooted atlas

A136572 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - 4\,x + 12\,n x + 16\,n^{2} x^{2}\right)\,P_{n}(x) = \left(n + 12\,n x + 48\,n x^{2} + 64\,n x^{3} + 12\,n^{2} x + 80\,n^{2} x^{2} + 128\,n^{2} x^{3} + 16\,n^{3} x^{2} + 64\,n^{3} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
111617
2264288354
36288259261449030
42415362073698304153600274200
512096001728001228800384000044236809675000
67206912015552001474560069120000159252480144506880389250000
7504056448015240960180633600112896000039016857607080837120528482304017592750000
840320516096016257024023121100801806336000083235962880226586787840338228674560214035333120882630000000
936288052254720188116992031213486080292626432000168552824832061178432716801369826131968017336861982720951268147200048676950000000
103628800580608000235146240004459069440004877107200000337105649664001529460817920004566087106560008668430991360009512681472000004604137832448002927137500000000
1139916800702535680031614105600067443425280008430428160000067982972682240037012951793664001381241349734400034962671665152000575517229056000005571006777262080024108939922636800190618312500000000
12479001600919683072004552431206400107909480448000151747706880000013985068666060800888310843047936003977975087235072001258656179945472000276248269946880000040111248796286976003471687348859699200135813694897520640013364331750000000000
13622702080012952203264006994189762560018236702195712002849484718080000029543457557053440021446361782157312001120462982904545280042542578882156953600116714894052556800000225960034885749964800293357580978644582400229525144376809881600819061052305047552001003681428750000000000
1487178291200195279372288001142384327884800324944875487232005584990047436800006433908534647193600525435863662854144003137296352132726784001389724243483793817600457522384686022656000011072041709401748275200191660286239381127168002249346414892736839680016053596625178932019200526539247910387712000080388770962500000000000
Technical details
Node id
n-fead27c2a1032954 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-6fb0656952daab5a

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:      1,       16;
  2:      2,       64,        288;
  3:      6,      288,       2592,        6144;
  4:     24,     1536,      20736,       98304,       153600;
  5:    120,     9600,     172800,     1228800,      3840000,       4423680;
  6:    720,    69120,    1555200,    14745600,     69120000,     159252480,     144506880;
  7:   5040,   564480,   15240960,   180633600,   1128960000,    3901685760,    7080837120,     5284823040;
  8:  40320,  5160960,  162570240,  2312110080,  18063360000,   83235962880,  226586787840,   338228674560,   214035333120;
  9: 362880, 52254720, 1881169920, 31213486080, 292626432000, 1685528248320, 6117843271680, 13698261319680, 17336861982720, 9512681472000;
%F Formula
The row polynomials satisfy (1+(12*n-4)*x+16*n^2*x^2)*P(n,x) = (n+(12*n^2+12*n)*x+(16*n^3+80*n^2+48*n)*x^2+(64*n^3+128*n^2+64*n)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n + (12*n^2 + 12*n)*x + (16*n^3 + 80*n^2 + 48*n)*x^2 + (64*n^3 + 128*n^2 + 64*n)*x^3)*P[n - 1, x])/(1 + (12*n - 4)*x + 16*n^2*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((n + (12*n^2 + 12*n)*x + (16*n^3 + 80*n^2 + 48*n)*x^2 + (64*n^3 + 128*n^2 + 64*n)*x^3)*v[i - 1])/(1 + (12*n - 4)*x + 16*n^2*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.