Real-rooted atlas

A038291 → basis A013609 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(5 + n x\right)\,P_{n}(x) = \left(50 + 20\,x + 2\,x^{2} + 10\,n x + 2\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
110414
21008012192
310001200360322592
41000016000720012808034560
5100000200000120000320004000192456192
6100000024000001800000640000120000115204485971968
710000000280000002520000011200000280000040320031360102477635584
8100000000320000000336000000179200000560000001075200012544008192023041003290624
910000000003600000000432000000026880000001008000000241920000376320003686400207360512012899450880
10100000000004000000000054000000000384000000001680000000048384000009408000001228800001036800051200011264165112971264
1110000000000044000000000066000000000052800000000026400000000088704000000206976000003379200000380160000281600001239040245762105190383616
12100000000000048000000000007920000000000704000000000039600000000001520640000000413952000000811008000001140480000011264000007434240029491205324826748301344768
13100000000000005200000000000093600000000000915200000000005720000000000024710400000000768768000000017571840000002965248000003660800000032215040001916928006922240114688338811817033728
141000000000000005600000000000001092000000000000116480000000000080080000000000038438400000000013453440000000035143680000000691891200000010250240000001127526400008945664000484556800160563202457604279728215162880
Technical details
Node id
n-ececab726f2e6291 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-6ae5c48b72ea3401

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:         10,          4;
  2:        100,         80,         12;
  3:       1000,       1200,        360,         32;
  4:      10000,      16000,       7200,       1280,         80;
  5:     100000,     200000,     120000,      32000,       4000,       192;
  6:    1000000,    2400000,    1800000,     640000,     120000,     11520,      448;
  7:   10000000,   28000000,   25200000,   11200000,    2800000,    403200,    31360,    1024;
  8:  100000000,  320000000,  336000000,  179200000,   56000000,  10752000,  1254400,   81920,   2304;
  9: 1000000000, 3600000000, 4320000000, 2688000000, 1008000000, 241920000, 37632000, 3686400, 207360, 5120;
%F Formula
The row polynomials satisfy (5+n*x)*P(n,x) = (50+(10*n+20)*x+(2*n+2)*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[((50 + (10*n + 20)*x + (2*n + 2)*x^2)*P[n - 1, x])/(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] = ((50 + (10*n + 20)*x + (2*n + 2)*x^2)*v[i - 1])/(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.