Real-rooted atlas

A027466 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(16 - 4\,x + 12\,n x + n^{2} x^{2}\right)\,P_{n}(x) = \left(128 + 96\,x + 24\,x^{2} + 2\,x^{3} + 96\,n x + 40\,n x^{2} + 4\,n x^{3} + 8\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
18816
26412836228
351215368641283040
440961638413824409640038800
53276816384018432081920160001152480000
62621441572864221184013107203840005529631365800000
7209715214680064247726081835008071680001548288175616819268800000
81677721613421772826424115223488102411468800033030144561971252428820736804000000
91342177281207959552271790899228185722881651507200594542592134873088188743681492992512009280000000
1010737418241073741824027179089920322122547202202009600095126814722697461760503316480597196804096000123904106000000000
1185899345929448928051226575110144035433480192027682406400013951932825647475326976110729625601751777280180224000109035522949121200000000000
126871947673682463372083225512105738243779571220480332188876800019134079303687596052316162126008811524204265472057671680005233704962831155269222413480000000000
13549755813888714682558054424120536334336393075406929923838627020800024874303094784112855634411523685081939968874487218176149946368000181435105281472200704719912961605632150400000000000
144398046511104615726511554562251250057871364002222325104644299262263296003095468829573121579978881761285896131103948816323761405952335879864320050801829478454962159616403151257617983078436864001668000000000000
Technical details
Node id
n-2e63cfc7b87a7680 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-9e3a89ee8f65d0f5

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:         8,          8;
  2:        64,        128,         36;
  3:       512,       1536,        864,        128;
  4:      4096,      16384,      13824,       4096,        400;
  5:     32768,     163840,     184320,      81920,      16000,      1152;
  6:    262144,    1572864,    2211840,    1310720,     384000,     55296,      3136;
  7:   2097152,   14680064,   24772608,   18350080,    7168000,   1548288,    175616,     8192;
  8:  16777216,  134217728,  264241152,  234881024,  114688000,  33030144,   5619712,   524288,   20736;
  9: 134217728, 1207959552, 2717908992, 2818572288, 1651507200, 594542592, 134873088, 18874368, 1492992, 51200;
%F Formula
The row polynomials satisfy (16+(12*n-4)*x+n^2*x^2)*P(n,x) = (128+(96*n+96)*x+(8*n^2+40*n+24)*x^2+(2*n^2+4*n+2)*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[((128 + (96*n + 96)*x + (8*n^2 + 40*n + 24)*x^2 + (2*n^2 + 4*n + 2)*x^3)*P[n - 1, x])/(16 + (12*n - 4)*x + n^2*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] = ((128 + (96*n + 96)*x + (8*n^2 + 40*n + 24)*x^2 + (2*n^2 + 4*n + 2)*x^3)*v[i - 1])/(16 + (12*n - 4)*x + n^2*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. [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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.