Real-rooted atlas

A178821 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4\,n - 2\,n x + 6\,n^{2} x + n^{3} x^{2}\right)\,P_{n}(x) = \left(32 + 48\,x + 24\,x^{2} + 4\,x^{3} + 8\,n + 60\,n x + 46\,n x^{2} + 9\,n x^{3} + 12\,n^{2} x + 18\,n^{2} x^{2} + 6\,n^{2} x^{3} + 2\,n^{3} x^{2} + n^{3} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1102030
260240135435
3280168018905604410
411208960151208960175035910
54032403209072080640315004536251748
61344016128045360053760031500090720102901581930
742240591360199584029568002310000997920226380211209141660
8126720202752079833601419264013860000798336027165605068804009549437135
9366080658944029652480615014407207200051891840235435206589440104247071500253320210
10102502420500480103783680246005760336336000290594304164804640615014401459458020020001211211241269029
11279552061501440345945600922521600144144000014529715209888278404612608001459458003003000036336301965605857070310
127454720178913280110702592032800768005765760000664215552052737484802952069120116756640032032000058138080628992030758026759825820
1319496960506920960342171648011152261120217817600002822916096025615349760167283916807939451520272272000065889824010692864010457720466480118893980520
145013504014037811201026514944036498309120784143360001129166438401152690739208603172864047636709120196035840005930084160128314368018823896016793280688500515508398820
Technical details
Node id
n-33410d3855a0f926 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-e1337d43fda52a72

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:     10,      20;
  2:     60,     240,      135;
  3:    280,    1680,     1890,      560;
  4:   1120,    8960,    15120,     8960,     1750;
  5:   4032,   40320,    90720,    80640,    31500,     4536;
  6:  13440,  161280,   453600,   537600,   315000,    90720,    10290;
  7:  42240,  591360,  1995840,  2956800,  2310000,   997920,   226380,   21120;
  8: 126720, 2027520,  7983360, 14192640, 13860000,  7983360,  2716560,  506880,   40095;
  9: 366080, 6589440, 29652480, 61501440, 72072000, 51891840, 23543520, 6589440, 1042470, 71500;
%F Formula
The row polynomials satisfy (4*n+(6*n^2-2*n)*x+n^3*x^2)*P(n,x) = (8*n+32+(12*n^2+60*n+48)*x+(2*n^3+18*n^2+46*n+24)*x^2+(n^3+6*n^2+9*n+4)*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[((8*n + 32 + (12*n^2 + 60*n + 48)*x + (2*n^3 + 18*n^2 + 46*n + 24)*x^2 + (n^3 + 6*n^2 + 9*n + 4)*x^3)*P[n - 1, x])/(4*n + (6*n^2 - 2*n)*x + 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] = ((8*n + 32 + (12*n^2 + 60*n + 48)*x + (2*n^3 + 18*n^2 + 46*n + 24)*x^2 + (n^3 + 6*n^2 + 9*n + 4)*x^3)*v[i - 1])/(4*n + (6*n^2 - 2*n)*x + 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.