Real-rooted atlas

A038291 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(25 - 5\,x + 15\,n x + n^{2} x^{2}\right)\,P_{n}(x) = \left(250 + 150\,x + 30\,x^{2} + 2\,x^{3} + 150\,n x + 50\,n x^{2} + 4\,n x^{3} + 10\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
110818
210016036296
31000240010801284608
4100003200021600512040069120
51000004000003600001280002000011521009152
6100000048000005400000256000060000069120313614432256
7100000005600000075600000448000001400000024192002195208192203046912

All 15 rows

Technical details
Node id
n-c4b5c836496940ed · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-7fe862d483d8a327

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:         10,          8;
  2:        100,        160,          36;
  3:       1000,       2400,        1080,         128;
  4:      10000,      32000,       21600,        5120,        400;
  5:     100000,     400000,      360000,      128000,      20000,       1152;
  6:    1000000,    4800000,     5400000,     2560000,     600000,      69120,      3136;
  7:   10000000,   56000000,    75600000,    44800000,   14000000,    2419200,    219520,     8192;
  8:  100000000,  640000000,  1008000000,   716800000,  280000000,   64512000,   8780800,   655360,   20736;
  9: 1000000000, 7200000000, 12960000000, 10752000000, 5040000000, 1451520000, 263424000, 29491200, 1866240, 51200;
%F Formula
The row polynomials satisfy (25+(15*n-5)*x+n^2*x^2)*P(n,x) = (250+(150*n+150)*x+(10*n^2+50*n+30)*x^2+(2*n^2+4*n+2)*x^3)*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[((250 + (150*n + 150)*x + (10*n^2 + 50*n + 30)*x^2 + (2*n^2 + 4*n + 2)*x^3)*P[n - 1, x])/(25 + (15*n - 5)*x + n^2*x^2)]];
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] = ((250 + (150*n + 150)*x + (10*n^2 + 50*n + 30)*x^2 + (2*n^2 + 4*n + 2)*x^3)*v[i - 1])/(25 + (15*n - 5)*x + n^2*x^2)); 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.