Real-rooted atlas

A135065 → basis A013609

palindromicγ-positivenonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(n^{2}\right)\,P_{n}(x) = \left(2 + 2\,x + 4\,n + 4\,n x + 2\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
18816
2367236144
31283843841281024
44001600240016004006400
51152576011520115205760115236864
6313618816470406272047040188163136200704
78192573441720322867202867201720325734481921048576
820736165888580608116121614515201161216580608165888207365308416
9512004608001843200430080064512006451200430080018432004608005120026214400
1012390412390405575680148684802601984031223808260198401486848055756801239040123904126877696
1129491232440321622016048660480973209601362493441362493449732096048660480162201603244032294912603979776
126922248306688456867841522892803426508805482414086396149765482414083426508801522892804568678483066886922242835349504
1316056322087321612523929645921075211480268802066448384275526451227552645122066448384114802688045921075212523929620873216160563213153337344
1436864005160960033546240013418496003690086400738017280011070259200126517248001107025920073801728003690086400134184960033546240051609600368640060397977600

Other recurrences

Image under the basis map A013609 · proved equal for all n (proof step)

\(P_{n}(x) = \left(4 + 8\,x + 4\,x^{2} + 16\,n + 32\,n x + 16\,n x^{2}\right)\,P_{n-2}(x) + \left(16 + 16\,x - 8\,n - 8\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 8 + 8\,x\)
Technical details
Node id
n-a24ba470fd16d40c · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A013609 (equal by e-0e7144cc4d5f72e2)
Fingerprint
f6-77a7ed3b1d5f257e

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: yes
  • γ-positive (gamma_positive): yes
  • 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:     8,      8;
  2:    36,     72,      36;
  3:   128,    384,     384,     128;
  4:   400,   1600,    2400,    1600,     400;
  5:  1152,   5760,   11520,   11520,    5760,    1152;
  6:  3136,  18816,   47040,   62720,   47040,   18816,    3136;
  7:  8192,  57344,  172032,  286720,  286720,  172032,   57344,    8192;
  8: 20736, 165888,  580608, 1161216, 1451520, 1161216,  580608,  165888,  20736;
  9: 51200, 460800, 1843200, 4300800, 6451200, 6451200, 4300800, 1843200, 460800, 51200;
%F Formula
The row polynomials satisfy n^2*P(n,x) = (2*n^2+4*n+2+(2*n^2+4*n+2)*x)*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[((2*n^2 + 4*n + 2 + (2*n^2 + 4*n + 2)*x)*P[n - 1, x])/(n^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] = ((2*n^2 + 4*n + 2 + (2*n^2 + 4*n + 2)*x)*v[i - 1])/(n^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. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [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
  7. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  8. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.