Real-rooted atlas

A109692 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-3 + 2\,n\right)\,P_{n}(x) = \left(4 + 36\,x + 81\,x^{2} - 16\,n - 132\,n x - 270\,n x^{2} + 20\,n^{2} + 144\,n^{2} x + 252\,n^{2} x^{2} - 8\,n^{3} - 48\,n^{3} x - 72\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(2 + 18\,x - 12\,n - 48\,n x + 8\,n^{2} + 24\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 5 + 3\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1538
2388427149
3348152418274054104
437922580059472501128505147681
5482884524001593612254134816677092296356532992
670617684890884078375298371368121021614653679727577955342317925
7116835841721318401054712304344706580863700817046411103992295098252329554024520713302000

All 15 rows

Technical details
Node id
n-3a7ef5c828a3b2c6 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-2eb18fe016005ab1

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:          5,           3;
  2:         38,          84,           27;
  3:        348,        1524,         1827,           405;
  4:       3792,       25800,        59472,         50112,           8505;
  5:      48288,      452400,      1593612,       2541348,        1667709,         229635;
  6:     706176,     8489088,     40783752,      98371368,      121021614,       65367972,        7577955;
  7:   11683584,   172131840,   1054712304,    3447065808,     6370081704,     6411103992,     2950982523,      295540245;
  8:  215967744,  3776601600,  28245930624,  117227590848,   292165637904,   439536802032,   375414436368,   150822168480,   13299311025;
  9: 4413837312, 89481120768, 792092805120, 4000466793600, 12619783474704, 25515760504464, 32459291153064, 24126746981688, 8609583053205, 678264862275;
%F Formula
The row polynomials satisfy (2*n-3)*P(n,x) = -(8*n^3-20*n^2+16*n-4+(48*n^3-144*n^2+132*n-36)*x+(72*n^3-252*n^2+270*n-81)*x^2)*P(n-2,x) + (8*n^2-12*n+2+(24*n^2-48*n+18)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 3*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 5 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(8*n^3 - 20*n^2 + 16*n - 4 + (48*n^3 - 144*n^2 + 132*n - 36)*x + (72*n^3 - 252*n^2 + 270*n - 81)*x^2)*P[n - 2, x] + (8*n^2 - 12*n + 2 + (24*n^2 - 48*n + 18)*x)*P[n - 1, x])/(2*n - 3)]];
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] = 5 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (-(8*n^3 - 20*n^2 + 16*n - 4 + (48*n^3 - 144*n^2 + 132*n - 36)*x + (72*n^3 - 252*n^2 + 270*n - 81)*x^2)*v[i - 2] + (8*n^2 - 12*n + 2 + (24*n^2 - 48*n + 18)*x)*v[i - 1])/(2*n - 3)); 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.