Real-rooted atlas

A038208 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4 - 2\,x + 6\,n x + n^{2} x^{2}\right)\,P_{n}(x) = \left(16 + 24\,x + 12\,x^{2} + 2\,x^{3} + 24\,n x + 20\,n x^{2} + 4\,n x^{3} + 4\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
14812
2166436116
3643844321281008
42562048345620484008208
510241024023040204808000115263936
640964915213824016384096000276483136482112
71638422937677414411468808960003870728780881923545856
8655361048576412876873400327168000412876814049282621442073625567488
926214447185922123366444040192516096003715891216859136471859274649651200181398528
10104857620971520106168320251658240344064000297271296168591360629145601492992020480001239041269789696
1141943049227468851904512013841203202162688000217998950414836039686920601602189721604505600054517762949128787750912
121677721640265318424914165767381975040129761280001494849945611868831744664377753626276659207208960001308426241415577669222460224311296
136710886417448304641177760563238386270208749731840009716524646488168464384575794053122732772556893716480002267938816368050176359956481605632409235079168
1426843545675161927685496215961619542101196841984983040060458375577661717925068846063524249625505877196810496245760031751143424687026995210078781448991539236864002760160002048
Technical details
Node id
n-87b168bcc9a83513 · 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:      4,       8;
  2:     16,      64,       36;
  3:     64,     384,      432,      128;
  4:    256,    2048,     3456,     2048,      400;
  5:   1024,   10240,    23040,    20480,     8000,     1152;
  6:   4096,   49152,   138240,   163840,    96000,    27648,     3136;
  7:  16384,  229376,   774144,  1146880,   896000,   387072,    87808,    8192;
  8:  65536, 1048576,  4128768,  7340032,  7168000,  4128768,  1404928,  262144,  20736;
  9: 262144, 4718592, 21233664, 44040192, 51609600, 37158912, 16859136, 4718592, 746496, 51200;
%F Formula
The row polynomials satisfy (4+(6*n-2)*x+n^2*x^2)*P(n,x) = (16+(24*n+24)*x+(4*n^2+20*n+12)*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[((16 + (24*n + 24)*x + (4*n^2 + 20*n + 12)*x^2 + (2*n^2 + 4*n + 2)*x^3)*P[n - 1, x])/(4 + (6*n - 2)*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] = ((16 + (24*n + 24)*x + (4*n^2 + 20*n + 12)*x^2 + (2*n^2 + 4*n + 2)*x^3)*v[i - 1])/(4 + (6*n - 2)*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.