Real-rooted atlas

A013613 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - 6\,x + 18\,n x + 36\,n^{2} x^{2}\right)\,P_{n}(x) = \left(1 + 18\,x + 108\,x^{2} + 216\,x^{3} + 18\,n x + 180\,n x^{2} + 432\,n x^{3} + 36\,n^{2} x^{2} + 216\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
112425
2148324373
317297234564501
41961944138243240048265
51120324034560162000279936479857
61144486069120486000167961622861444525885
71168680412096011340005878656160030081791590441059501
81192907219353622680001567641664012032143327232136048896361535377
9121611664290304408240035271936192036096644972544122444006410077696003108874825
101240145804147206804000705438724800902402149908480612220032010077696000731640729626224079749
11126417820570240106920001293304321056198528591224832022448067840554273280008048048025652242776064217707709765
1212882138476032016038000221709312211239705614189395968673442035202217093120004828828815366269133127683678762147841783266246937
1313122527298841623166000360277632392302310430743691264175094929152720555264000209249248665640749365329924782390792192255989602713614440417204129
141336294841257984324324005604318726865290432614873825284085548346882017554739200732372370329619016370487296334767355453443583854437990417631936921600115782367436365
Technical details
Node id
n-eb131f6345810628 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-544ecf665d341b3c

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: 1,  24;
  2: 1,  48,   324;
  3: 1,  72,   972,   3456;
  4: 1,  96,  1944,  13824,   32400;
  5: 1, 120,  3240,  34560,  162000,   279936;
  6: 1, 144,  4860,  69120,  486000,  1679616,   2286144;
  7: 1, 168,  6804, 120960, 1134000,  5878656,  16003008,  17915904;
  8: 1, 192,  9072, 193536, 2268000, 15676416,  64012032, 143327232,  136048896;
  9: 1, 216, 11664, 290304, 4082400, 35271936, 192036096, 644972544, 1224440064, 1007769600;
%F Formula
The row polynomials satisfy (1+(18*n-6)*x+36*n^2*x^2)*P(n,x) = (1+(18*n+18)*x+(36*n^2+180*n+108)*x^2+(216*n^2+432*n+216)*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[((1 + (18*n + 18)*x + (36*n^2 + 180*n + 108)*x^2 + (216*n^2 + 432*n + 216)*x^3)*P[n - 1, x])/(1 + (18*n - 6)*x + 36*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] = ((1 + (18*n + 18)*x + (36*n^2 + 180*n + 108)*x^2 + (216*n^2 + 432*n + 216)*x^3)*v[i - 1])/(1 + (18*n - 6)*x + 36*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

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.