Real-rooted atlas

A038284 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(49 + 315\,x + 324\,x^{2} + 189\,n x + 324\,n x^{2} + 81\,n^{2} x^{2}\right)\,P_{n}(x) = \left(686 + 7938\,x + 19278\,x^{2} + 13122\,x^{3} + 2646\,n x + 10206\,n x^{2} + 8748\,n x^{3} + 1134\,n^{2} x^{2} + 1458\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 196 + 2016\,x + 2916\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0196201629165128
127444233612247293312260864
23841679027234292165225472262440012107776
3537824138297608001504018289152018370800068024448529006592
475295362323399681680315840512096256077157360005714053632166659897622137536512
510541350437948861443293419046412546358272025204737600027998862796816332669964839182082048896842858496
614757890566071817830461477155532828103842529287057326528000104529087774729146295180288438839318937689261680665635424890257408
7206610467849563113082881106588799590459018069311488177844628505600329266626490368384144397572096276468770930688112469717638656198359290368001371090999836672

All 15 rows

Technical details
Node id
n-e8a5f7233da60984 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-238fadf8fd88f70e

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:           196,            2016,             2916;
  1:          2744,           42336,           122472,             93312;
  2:         38416,          790272,          3429216,           5225472,           2624400;
  3:        537824,        13829760,         80015040,         182891520,         183708000,           68024448;
  4:       7529536,       232339968,       1680315840,        5120962560,        7715736000,         5714053632,         1666598976;
  5:     105413504,      3794886144,      32934190464,      125463582720,      252047376000,       279988627968,       163326699648,        39182082048;
  6:    1475789056,     60718178304,     614771555328,     2810384252928,     7057326528000,     10452908777472,      9146295180288,      4388393189376,       892616806656;
  7:   20661046784,    956311308288,   11065887995904,    59018069311488,   177844628505600,    329266626490368,    384144397572096,    276468770930688,    112469717638656,     19835929036800;
  8:  289254654976,  14875953684480,  193653039928320,  1180361386229760,  4149707998464000,   9219465541730304,  13445053915023360,  12901875976765440,   7872880234705920,   2777030065152000,   432026534421504;
  9: 4049565169664, 229089686740992, 3313618683217920, 22721956684922880, 91293575966208000, 236632948904411136, 414107660582719488, 496722225105469440, 404141185381570560, 213831315016704000, 66532086300911616, 9254651051409408;
%F Formula
The row polynomials satisfy (49+(189*n+315)*x+(81*n^2+324*n+324)*x^2)*P(n,x) = (686+(2646*n+7938)*x+(1134*n^2+10206*n+19278)*x^2+(1458*n^2+8748*n+13122)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 196 + 2016*x + 2916*x^2.
%t Mathematica
P[0, x_] := 196 + 2016*x + 2916*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((686 + (2646*n + 7938)*x + (1134*n^2 + 10206*n + 19278)*x^2 + (1458*n^2 + 8748*n + 13122)*x^3)*P[n - 1, x])/(49 + (189*n + 315)*x + (81*n^2 + 324*n + 324)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 196 + 2016*x + 2916*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((686 + (2646*n + 7938)*x + (1134*n^2 + 10206*n + 19278)*x^2 + (1458*n^2 + 8748*n + 13122)*x^3)*v[i - 1])/(49 + (189*n + 315)*x + (81*n^2 + 324*n + 324)*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.