Real-rooted atlas

A143495 → D+1 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(5 + x\right)\,P_{n-1}(x) + \left(x\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2516
32511137
412591181235
56256712172611540
631254651219042535110427
7156253103119981559074045173013
87812520181117089864701122501190561529032
9390625128899113980976885061749512415018066813967195
1019531258124571110756706906145226306541645144016262281130785747
119765625507005518565426166324830272737306427575900627754803675951247126450
124882812531396893165028037861625290131178840091549480163344721806387123255500511012050937445
13244140625193264171148659315775580303586342234850112272832002712286723801111634087021933256655126117585497797
1412207031251183999089135994162750495083602653638144009515695180501421079859272736206482324242611525229315086711431155666739742
1561035156257226064847126379913014143206104487037694132112019333324510562013965013129391433601797577210167937770105125024318861110216111421428484337
Technical details
Node id
n-67748ed3490e4da5 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-73b18cc810b5ad72

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:       1;
   2:       5,       1;
   3:      25,      11,        1;
   4:     125,      91,       18,       1;
   5:     625,     671,      217,      26,       1;
   6:    3125,    4651,     2190,     425,      35,      1;
   7:   15625,   31031,    19981,    5590,     740,     45,     1;
   8:   78125,  201811,   170898,   64701,   12250,   1190,    56,    1;
   9:  390625, 1288991,  1398097,  688506,  174951,  24150,  1806,   68,  1;
  10: 1953125, 8124571, 11075670, 6906145, 2263065, 416451, 44016, 2622, 81, 1;
%F Formula
The row polynomials satisfy P(n,x) = (5+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(5 + x)*P[n - 1, x] + x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (5 + x)*v[i - 1] + x*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.