Real-rooted atlas

A038208 → basis A007318 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(2 + n x\right)\,P_{n}(x) = \left(8 + 8\,x + 2\,x^{2} + 4\,n x + 2\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1448
216321260
36419214432432
425610241152512803024
51024512076805120160019220736
64096245764608040960192004608448139968
71638411468825804828672017920064512125441024933120
8655365242881376256183500814336006881282007043276823046158592
92621442359296707788811010048103219206193152240844858982482944512040310784
1010485761048576035389440629145606881280049545216240844807864320165888020480011264262020096
1141943044613734417301504034603008043253760036333158421194342486507520243302404505600495616245761693052928
12167772162013265928304721921845493760259522560024914165761695547392830472192291962880720896001189478411796485324810883911680
1367108864872415232392586854495965675521499463680016194207744125954949127197425664303641395293716480020617625630670848276889611468869657034752
142684354563758096384183207198724885525299283969966080100763959296881684643845757940531228339863552104962457602886467584572522496775290886422528245760444063596544
Technical details
Node id
n-d03b3ca3ebd61dd6 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-e620d4a356abb37c

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:      4,       4;
  2:     16,      32,      12;
  3:     64,     192,     144,       32;
  4:    256,    1024,    1152,      512,       80;
  5:   1024,    5120,    7680,     5120,     1600,     192;
  6:   4096,   24576,   46080,    40960,    19200,    4608,     448;
  7:  16384,  114688,  258048,   286720,   179200,   64512,   12544,   1024;
  8:  65536,  524288, 1376256,  1835008,  1433600,  688128,  200704,  32768,  2304;
  9: 262144, 2359296, 7077888, 11010048, 10321920, 6193152, 2408448, 589824, 82944, 5120;
%F Formula
The row polynomials satisfy (2+n*x)*P(n,x) = (8+(4*n+8)*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[((8 + (4*n + 8)*x + (2*n + 2)*x^2)*P[n - 1, x])/(2 + 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] = ((8 + (4*n + 8)*x + (2*n + 2)*x^2)*v[i - 1])/(2 + 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.