Real-rooted atlas

A013622 → basis A013609 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(4 + 5\,n x\right)\,P_{n}(x) = \left(32 + 80\,x + 50\,x^{2} + 40\,n x + 50\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
182028
264320300684
351238407200400015552
440964096011520012800050000338256
5327684096001536000256000020000006000007138368
62621443932160184320004096000048000000288000007000000147386304
7209715236700160206438400573440000896000000806400000392000000800000002993075712

All 15 rows

Technical details
Node id
n-280e93fc5b9e9fe7 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-eefdbaf8408bdfd8

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:         8,         20;
  2:        64,        320,         300;
  3:       512,       3840,        7200,        4000;
  4:      4096,      40960,      115200,      128000,        50000;
  5:     32768,     409600,     1536000,     2560000,      2000000,       600000;
  6:    262144,    3932160,    18432000,    40960000,     48000000,     28800000,      7000000;
  7:   2097152,   36700160,   206438400,   573440000,    896000000,    806400000,    392000000,     80000000;
  8:  16777216,  335544320,  2202009600,  7340032000,  14336000000,  17203200000,  12544000000,   5120000000,   900000000;
  9: 134217728, 3019898880, 22649241600, 88080384000, 206438400000, 309657600000, 301056000000, 184320000000, 64800000000, 10000000000;
%F Formula
The row polynomials satisfy (4+5*n*x)*P(n,x) = (32+(40*n+80)*x+(50*n+50)*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[((32 + (40*n + 80)*x + (50*n + 50)*x^2)*P[n - 1, x])/(4 + 5*n*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] = ((32 + (40*n + 80)*x + (50*n + 50)*x^2)*v[i - 1])/(4 + 5*n*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. [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.