Real-rooted atlas

A038338 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4 + 10\,x + 4\,x^{2} + 6\,n x + 4\,n x^{2} + n^{2} x^{2}\right)\,P_{n}(x) = \left(96 + 432\,x + 408\,x^{2} + 108\,x^{3} + 144\,n x + 216\,n x^{2} + 72\,n x^{3} + 24\,n^{2} x^{2} + 12\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 576 + 2304\,x + 1296\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0576230412964176
113824829449331227648217728
233177626542084478976265420851840010637568
3796262479626240179159040159252480622080008957952497166336
41911029762293235712644972544076441190404478976000128994508814631321622493417472
5458647142464210599936216710774784321052999680250822656000108355387392245806202882293235712992612745216
6110075314176176120502681669347447930881232843518771212039487488000693474479308823597395476484403012567043482851737642943561924608
72641807540224475525357240322139864107581444438236667576325201058594816003744762188267521699012474306564755253572403275229597532165159780352001828079220031488

All 15 rows

Technical details
Node id
n-c5f7f3ac90e6faee · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-e1337d43fda52a72

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,              2304,               1296;
  1:            13824,             82944,              93312,              27648;
  2:           331776,           2654208,            4478976,            2654208,             518400;
  3:          7962624,          79626240,          179159040,          159252480,           62208000,            8957952;
  4:        191102976,        2293235712,         6449725440,         7644119040,         4478976000,         1289945088,          146313216;
  5:       4586471424,       64210599936,       216710774784,       321052999680,       250822656000,       108355387392,        24580620288,         2293235712;
  6:     110075314176,     1761205026816,      6934744793088,     12328435187712,     12039487488000,      6934744793088,      2359739547648,       440301256704,       34828517376;
  7:    2641807540224,    47552535724032,    213986410758144,    443823666757632,    520105859481600,    374476218826752,    169901247430656,     47552535724032,     7522959753216,      515978035200;
  8:   63403380965376,  1268067619307520,   6419592322744320,  15216811431690240,  20804234379264000,  17974858503684096,  10194074845839360,   3804202857922560,   902755170385920,   123834728448000,    7492001071104;
  9: 1521681143169024, 33476985149718528, 188308041467166720, 502154777245777920, 784616839446528000, 790893774162100224, 538247151860318208, 251077388622888960, 79442454993960960, 16346184155136000, 1977888282771456, 106993205379072;
%F Formula
The row polynomials satisfy (4+(6*n+10)*x+(n^2+4*n+4)*x^2)*P(n,x) = (96+(144*n+432)*x+(24*n^2+216*n+408)*x^2+(12*n^2+72*n+108)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 576 + 2304*x + 1296*x^2.
%t Mathematica
P[0, x_] := 576 + 2304*x + 1296*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((96 + (144*n + 432)*x + (24*n^2 + 216*n + 408)*x^2 + (12*n^2 + 72*n + 108)*x^3)*P[n - 1, x])/(4 + (6*n + 10)*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 + 2304*x + 1296*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((96 + (144*n + 432)*x + (24*n^2 + 216*n + 408)*x^2 + (12*n^2 + 72*n + 108)*x^3)*v[i - 1])/(4 + (6*n + 10)*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.