Real-rooted atlas

A178821 → basis A013610 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(2\,n + 3\,n^{2} x\right)\,P_{n}(x) = \left(16 + 48\,x + 36\,x^{2} + 4\,n + 36\,n x + 45\,n x^{2} + 6\,n^{2} x + 9\,n^{2} x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1103040
260360405825
328025205670378012250
4112013440453606048028350148750
54032604802721605443205103001837081575000
6134402419201360800362880051030003674160107163015093750
7422408870405987520199584003742200040415760235758605773680134062500

All 15 rows

Technical details
Node id
n-3e2fae9e466027e9 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-311883d116e971c2

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      1;
  1:     10,      30;
  2:     60,     360,      405;
  3:    280,    2520,     5670,      3780;
  4:   1120,   13440,    45360,     60480,      28350;
  5:   4032,   60480,   272160,    544320,     510300,     183708;
  6:  13440,  241920,  1360800,   3628800,    5103000,    3674160,    1071630;
  7:  42240,  887040,  5987520,  19958400,   37422000,   40415760,   23575860,    5773680;
  8: 126720, 3041280, 23950080,  95800320,  224532000,  323326080,  282910320,  138568320,  29229255;
  9: 366080, 9884160, 88957440, 415134720, 1167566400, 2101619520, 2451889440, 1801388160, 759960630, 140733450;
%F Formula
The row polynomials satisfy (2*n+3*n^2*x)*P(n,x) = (4*n+16+(6*n^2+36*n+48)*x+(9*n^2+45*n+36)*x^2)*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[((4*n + 16 + (6*n^2 + 36*n + 48)*x + (9*n^2 + 45*n + 36)*x^2)*P[n - 1, x])/(2*n + 3*n^2*x)]];
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] = ((4*n + 16 + (6*n^2 + 36*n + 48)*x + (9*n^2 + 45*n + 36)*x^2)*v[i - 1])/(2*n + 3*n^2*x)); 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.