Real-rooted atlas

A094638 → Veronese 2 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(n x - 2\,n^{2} - 2\,n^{2} x + n^{3} + n^{3} x\right)\,P_{n-2}(x) + \left(-1 + 2\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111
2325
3121123
4608324167
53606332741267
625205583378472012607
720160530644597313068132265
8181440561096618741279404403201681001
91814400640476083937694829984102657622469489
10199584007972596012321801990719234308975763628800348147989
112395008001064486880187736288416398102397079772761205438405649681919
123113510400152971070403057958069231569445095175000285964805569152479001600103344242575
1343589145600234137856960520602213168612672223377401946269866139130307336198027590401971880775347
1465383718400038205448848009397876061520125495979086079746609510688434340650782410024048362248717829120041601455184863
Technical details
Node id
n-37435152a1be1ee6 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-dda871f0e8e6411f

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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:       1;
  2:       3,       2;
  3:      12,      11;
  4:      60,      83,      24;
  5:     360,     633,     274;
  6:    2520,    5583,    3784,     720;
  7:   20160,   53064,   45973,   13068;
  8:  181440,  561096,  618741,  279404,   40320;
  9: 1814400, 6404760, 8393769, 4829984, 1026576;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (n^3-2*n^2+(n^3-2*n^2+n)*x)*P(n-2,x) + (2*n-1)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^3 - 2*n^2 + (n^3 - 2*n^2 + n)*x)*P[n - 2, x] + (2*n - 1)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((n^3 - 2*n^2 + (n^3 - 2*n^2 + n)*x)*v[i - 2] + (2*n - 1)*v[i - 1])/(n - 1)); 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.