Real-rooted atlas

A178822 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(n - 2\,n x + 6\,n^{2} x + 4\,n^{3} x^{2}\right)\,P_{n}(x) = \left(10 + 60\,x + 120\,x^{2} + 80\,x^{3} + 2\,n + 72\,n x + 224\,n x^{2} + 176\,n x^{3} + 12\,n^{2} x + 80\,n^{2} x^{2} + 112\,n^{2} x^{3} + 8\,n^{3} x^{2} + 16\,n^{3} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
11296108
284134430244452
3448107524838457344116928
420166451243545610321928064002340576
5806432256029030401032192016128000928972838973312
6295681419264159667207569408017740800020437401692725248567616896
71013765677056766402564541644801419264000245248819222254059528304721927464213504
83294722108620833210777623616552969225216000212548976642893027737621592276992683193139290549778176
910250247380172813284311041102105804851661209600148784283648270015922176302291877888191294078976524812288001028952916992
103075072246005760498161664047233105920258306048000892705701888202511941632030229187788802869411184640157443686400038101372108811076375518208
118945664787218432177124147201889324236801180827648000476109707673612960764264448241833502310403060705263616025190989824000121924390748162638183661568113922145419264
12253460482433220608602222100487137447116805018517504000231253286584327344433083187216444678157107226015994740736028549788467200020727146427187289698244493312175451427307521126984067629056
13701890567299661824197090869248256948096204820074070016000104063978962944377713701421056986680689426432187315162133299225694809620480002487257571262464161456840087961663162513830707211269779436339210780087869702144
14190513152213374730246241210859528876388777984762814660608004393812445102081794140081750016535626659974348811863293601775616195280553115648002362894692699340820451199744475136120008776278343684282516185808896702307683532800100132788511899648
Technical details
Node id
n-478b0ced7e7af25c · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-181bf1738e48841f

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:      12,       96;
  2:      84,     1344,       3024;
  3:     448,    10752,      48384,       57344;
  4:    2016,    64512,     435456,     1032192,      806400;
  5:    8064,   322560,    2903040,    10321920,    16128000,      9289728;
  6:   29568,  1419264,   15966720,    75694080,   177408000,    204374016,     92725248;
  7:  101376,  5677056,   76640256,   454164480,  1419264000,   2452488192,   2225405952,    830472192;
  8:  329472, 21086208,  332107776,  2361655296,  9225216000,  21254897664,  28930277376,  21592276992,   6831931392;
  9: 1025024, 73801728, 1328431104, 11021058048, 51661209600, 148784283648, 270015922176, 302291877888, 191294078976, 52481228800;
%F Formula
The row polynomials satisfy (n+(6*n^2-2*n)*x+4*n^3*x^2)*P(n,x) = (2*n+10+(12*n^2+72*n+60)*x+(8*n^3+80*n^2+224*n+120)*x^2+(16*n^3+112*n^2+176*n+80)*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[((2*n + 10 + (12*n^2 + 72*n + 60)*x + (8*n^3 + 80*n^2 + 224*n + 120)*x^2 + (16*n^3 + 112*n^2 + 176*n + 80)*x^3)*P[n - 1, x])/(n + (6*n^2 - 2*n)*x + 4*n^3*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] = ((2*n + 10 + (12*n^2 + 72*n + 60)*x + (8*n^3 + 80*n^2 + 224*n + 120)*x^2 + (16*n^3 + 112*n^2 + 176*n + 80)*x^3)*v[i - 1])/(n + (6*n^2 - 2*n)*x + 4*n^3*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.