Real-rooted atlas

A038335 → basis A013609 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(7 + 6\,n x\right)\,P_{n}(x) = \left(147 + 252\,x + 108\,x^{2} + 126\,n x + 108\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1213657
244115129722925
39261476286123623328141453
41944811333584257191219595525248806584409
5408410135006580900169201028764805511240011337408298433889
685766121882165816283553298043208121603472081200142851340823808556813262957253
718010885412161306249283364669612158789846880170131978800104995735488349985784964897760256580592720565

All 15 rows

Technical details
Node id
n-cbc748e5c1bed4d4 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-b92d3759e52bf2f7

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:           21,             36;
  2:          441,           1512,            972;
  3:         9261,          47628,          61236,           23328;
  4:       194481,        1333584,        2571912,         1959552,          524880;
  5:      4084101,       35006580,       90016920,       102876480,        55112400,        11337408;
  6:     85766121,      882165816,     2835532980,      4320812160,      3472081200,      1428513408,       238085568;
  7:   1801088541,    21613062492,    83364669612,    158789846880,    170131978800,    104995735488,     34998578496,     4897760256;
  8:  37822859361,   518713499808,  2334210749136,   5335338855168,   7145543109600,   5879761187328,   2939880593664,   822823723008,    99179645184;
  9: 794280046581, 12254606432964, 63023690226672, 168063173937792, 270101529542880, 277818716101248, 185212477400832, 77756841824256, 18744952939776, 1983592903680;
%F Formula
The row polynomials satisfy (7+6*n*x)*P(n,x) = (147+(126*n+252)*x+(108*n+108)*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[((147 + (126*n + 252)*x + (108*n + 108)*x^2)*P[n - 1, x])/(7 + 6*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] = ((147 + (126*n + 252)*x + (108*n + 108)*x^2)*v[i - 1])/(7 + 6*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.