Real-rooted atlas

A038338 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4 + 30\,x + 36\,x^{2} + 18\,n x + 36\,n x^{2} + 9\,n^{2} x^{2}\right)\,P_{n}(x) = \left(96 + 1296\,x + 3672\,x^{2} + 2916\,x^{3} + 432\,n x + 1944\,n x^{2} + 1944\,n x^{3} + 216\,n^{2} x^{2} + 324\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 576 + 6912\,x + 11664\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
057669121166419152
1138242488328398087464961848960
23317767962624403107847166361641990400162259200
37962624238878720161243136042998169605038848000217678233613374720000
41911029766879707136580475289602063912140803627970560003134566563841066623344641054425600000
5458647142419263179980819503969730568668430991360203166351360002633035913625617919272189952501530650214480397619200000
6110075314176528361508044862412703137792332867750068224975198486528000168514298472038417202501302353929629388484116482285099025039365972714496000000
7264180754022414265760717209619258776968232961198323900245606442128574618009600909977211749007361238580093769482241039973956284579844935813894085017610155995666841600434550251520000000

All 15 rows

Technical details
Node id
n-25826dc4ec8242bd · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-56805ed74e931bf6

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,               6912,               11664;
  1:            13824,             248832,              839808,               746496;
  2:           331776,            7962624,            40310784,             71663616,             41990400;
  3:          7962624,          238878720,          1612431360,           4299816960,           5038848000,            2176782336;
  4:        191102976,         6879707136,         58047528960,         206391214080,         362797056000,          313456656384,          106662334464;
  5:       4586471424,       192631799808,       1950396973056,        8668430991360,       20316635136000,        26330359136256,        17919272189952,         5015306502144;
  6:     110075314176,      5283615080448,      62412703137792,      332867750068224,      975198486528000,      1685142984720384,      1720250130235392,       962938848411648,       228509902503936;
  7:    2641807540224,    142657607172096,    1925877696823296,    11983239002456064,    42128574618009600,     90997721174900736,    123858009376948224,    103997395628457984,     49358138940850176,     10155995666841600;
  8:   63403380965376,   3804202857922560,   57776330904698880,   410853908655636480,  1685142984720384000,   4367890616395235328,   7431480562616893440,   8319791650276638720,   5922976672902021120,   2437438960041984000,    442395171247620096;
  9: 1521681143169024, 100430955449155584, 1694772373204500480, 13558178985636003840, 63553963995168768000, 192187187121390354432, 392382173706171973632, 549106248918258155520, 521221947215377858560, 321741942725541888000, 116792325209371705344, 18953525353286467584;
%F Formula
The row polynomials satisfy (4+(18*n+30)*x+(9*n^2+36*n+36)*x^2)*P(n,x) = (96+(432*n+1296)*x+(216*n^2+1944*n+3672)*x^2+(324*n^2+1944*n+2916)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 576 + 6912*x + 11664*x^2.
%t Mathematica
P[0, x_] := 576 + 6912*x + 11664*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((96 + (432*n + 1296)*x + (216*n^2 + 1944*n + 3672)*x^2 + (324*n^2 + 1944*n + 2916)*x^3)*P[n - 1, x])/(4 + (18*n + 30)*x + (9*n^2 + 36*n + 36)*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 + 6912*x + 11664*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((96 + (432*n + 1296)*x + (216*n^2 + 1944*n + 3672)*x^2 + (324*n^2 + 1944*n + 2916)*x^3)*v[i - 1])/(4 + (18*n + 30)*x + (9*n^2 + 36*n + 36)*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.