Real-rooted atlas

A094638 → D+1 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-4 + 8\,n - 5\,n^{2} + n^{3}\right)\,P_{n}(x) = \left(n - 6\,n x + 12\,n x^{2} - 8\,n x^{3} - 2\,n^{2} + 15\,n^{2} x - 36\,n^{2} x^{2} + 28\,n^{2} x^{3} + n^{3} - 12\,n^{3} x + 39\,n^{3} x^{2} - 38\,n^{3} x^{3} + 3\,n^{4} x - 18\,n^{4} x^{2} + 25\,n^{4} x^{3} + 3\,n^{5} x^{2} - 8\,n^{5} x^{3} + n^{6} x^{3}\right)\,P_{n-3}(x) + \left(-7\,n + 18\,n x - 12\,n x^{2} + 9\,n^{2} - 39\,n^{2} x + 36\,n^{2} x^{2} - 3\,n^{3} + 27\,n^{3} x - 39\,n^{3} x^{2} - 6\,n^{4} x + 18\,n^{4} x^{2} - 3\,n^{5} x^{2}\right)\,P_{n-2}(x) + \left(-4 + 14\,n - 12\,n x - 12\,n^{2} + 24\,n^{2} x + 3\,n^{3} - 15\,n^{3} x + 3\,n^{4} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 2\,x\)
\(P_{2}(x) = 4 + 14\,x + 6\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1224
2414624
37568724174
4111605555841201430
51637022805284437072013040
62274271402892452220365045040130592
729134418606116144362145549240338436403201426264
837225642462377328179434546561686196484345715236288016889112
94635708766010501687049490276684306252806075082464386324643628800215731152
10565390167310259696823340240128421810435984010883504864975702024469166400399168002958805872
11677832299805584944867726065496048608234716751571044827041317167874013563563760615591504047900160043391741184
1279110245100811221448817691316516612081201041236896743709798704120450067452207296104080201106743552868135449606227020800677866505472
1392151068310122395593242411369049613309463970146980821983914752883659907331021309523471603441654047832317075790643213098023827208717829120011241894912768
141062023013049404456925294655561013486691790133970116280943648297592471941661391216538794235600393866381708806020245001145653003732775552210554641382401307674368000197306267869440
Technical details
Node id
n-c31feacfee4079b7 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-a45348decf901a8a

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:  2,    2;
  2:  4,   14,     6;
  3:  7,   56,    87,      24;
  4: 11,  160,   555,     584,     120;
  5: 16,  370,  2280,    5284,    4370,      720;
  6: 22,  742,  7140,   28924,   52220,    36504,     5040;
  7: 29, 1344, 18606,  116144,  362145,   549240,   338436,    40320;
  8: 37, 2256, 42462,  377328, 1794345,  4656168,  6196484,  3457152,   362880;
  9: 46, 3570, 87660, 1050168, 7049490, 27668430, 62528060, 75082464, 38632464, 3628800;
%F Formula
The row polynomials satisfy (n^3-5*n^2+8*n-4)*P(n,x) = (n^3-2*n^2+n+(3*n^4-12*n^3+15*n^2-6*n)*x+(3*n^5-18*n^4+39*n^3-36*n^2+12*n)*x^2+(n^6-8*n^5+25*n^4-38*n^3+28*n^2-8*n)*x^3)*P(n-3,x) - (3*n^3-9*n^2+7*n+(6*n^4-27*n^3+39*n^2-18*n)*x+(3*n^5-18*n^4+39*n^3-36*n^2+12*n)*x^2)*P(n-2,x) + (3*n^3-12*n^2+14*n-4+(3*n^4-15*n^3+24*n^2-12*n)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2 + 2*x, P(2,x) = 4 + 14*x + 6*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[2, x_] := 4 + 14*x + 6*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n^3 - 2*n^2 + n + (3*n^4 - 12*n^3 + 15*n^2 - 6*n)*x + (3*n^5 - 18*n^4 + 39*n^3 - 36*n^2 + 12*n)*x^2 + (n^6 - 8*n^5 + 25*n^4 - 38*n^3 + 28*n^2 - 8*n)*x^3)*P[n - 3, x] - (3*n^3 - 9*n^2 + 7*n + (6*n^4 - 27*n^3 + 39*n^2 - 18*n)*x + (3*n^5 - 18*n^4 + 39*n^3 - 36*n^2 + 12*n)*x^2)*P[n - 2, x] + (3*n^3 - 12*n^2 + 14*n - 4 + (3*n^4 - 15*n^3 + 24*n^2 - 12*n)*x)*P[n - 1, x])/(n^3 - 5*n^2 + 8*n - 4)]];
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] = 2 + 2*x; v[3] = 4 + 14*x + 6*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n^3 - 2*n^2 + n + (3*n^4 - 12*n^3 + 15*n^2 - 6*n)*x + (3*n^5 - 18*n^4 + 39*n^3 - 36*n^2 + 12*n)*x^2 + (n^6 - 8*n^5 + 25*n^4 - 38*n^3 + 28*n^2 - 8*n)*x^3)*v[i - 3] - (3*n^3 - 9*n^2 + 7*n + (6*n^4 - 27*n^3 + 39*n^2 - 18*n)*x + (3*n^5 - 18*n^4 + 39*n^3 - 36*n^2 + 12*n)*x^2)*v[i - 2] + (3*n^3 - 12*n^2 + 14*n - 4 + (3*n^4 - 15*n^3 + 24*n^2 - 12*n)*x)*v[i - 1])/(n^3 - 5*n^2 + 8*n - 4)); 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. [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
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.