Real-rooted atlas

A331333 → D+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1325
21016430
342108608218
4216768672192161864
513206000720032005603218312
6936051840792004800012960153664202960
7756004939209172807056002587204704040321282502320
868544051609601128960010536960489216012042241576961024025633937536
96894720587865601480550401625702409144576028449792503193649766425344512501757568
10762048007257600002068416000261273600017273088006502809601451520001935360014976006144010248026772224
1191808640096598656003073593600043908480000333704448001475324928039810355206690816006969600043366401464322048138070360320
12119750400001379524608004847496192007727892480006638962176003372171264001067854233602167824384028435968002378956801216512034406440962540137384960
13168129561600210473303040080951270400001424742359040013653780940800783608297472028494847180806784487424001074210508801130747904077667532833226752798720819249752749837312
1425281704448003417389015040014279804098560027501844930560029088489830400018616633491456076461173268480208962212659203878465310720492503531520425343959042442166272887111681835008163841033343214606336
Technical details
Node id
n-b8e14b82713bc7db · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-fefd6e20ffcdfb8c

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:       3,        2;
  2:      10,       16,         4;
  3:      42,      108,        60,         8;
  4:     216,      768,       672,       192,       16;
  5:    1320,     6000,      7200,      3200,      560,       32;
  6:    9360,    51840,     79200,     48000,    12960,     1536,      64;
  7:   75600,   493920,    917280,    705600,   258720,    47040,    4032,    128;
  8:  685440,  5160960,  11289600,  10536960,  4892160,  1204224,  157696,  10240,   256;
  9: 6894720, 58786560, 148055040, 162570240, 91445760, 28449792, 5031936, 497664, 25344, 512;
%F Formula
The row polynomials satisfy (2*n-1+x)*P(n,x) = (2*n^2+n+5*n*x+2*x^2)*P(n-1,x) + (2*n*x+x^2)*d/dx 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[((2*n^2 + n + 5*n*x + 2*x^2)*P[n - 1, x] + (2*n*x + x^2)*D[P[n - 1, x], x])/(2*n - 1 + 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] = ((2*n^2 + n + 5*n*x + 2*x^2)*v[i - 1] + (2*n*x + x^2)*deriv(v[i - 1], x))/(2*n - 1 + 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

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.