Real-rooted atlas

A136124 → basis A008277 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(1 + n x\right)\,P_{n}(x) = \left(2 + 2\,x + x^{2} + n + 2\,n x + n x^{2} + n^{2} x\right)\,P_{n-1}(x) + \left(x + x^{2} + n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1325
21216331
360120454229
43609605409651961
52520840063001680175619081
62016080640756002688042002887207775
718144084672095256042336088200907244182501801
818144009676800127008006773760176400024192017640640932989969
9199584001197504001796256001117670403492720059875205821203168089110472630861
10239500800159667200026943840001916006400698544000143700480174636001267200534601200117307593151
11311351040022832409600428107680003424861440014270256000342486144049945896045302400254826085800157312121247816845
12435891456003487131648007192209024006393074688002996753760008219667456013984850880152216064010702692048048001321322016132148321709561
136538371840005666588928000127498250880001246649564160064929664800002003543942400389577988800494702208004174049880234234000858858019656025351440476722545169
141046139494400097639686144000237996734976000253863183974400145442449152000498659825664001090818368640015830470656001558311955201049368320048096048014676480283920313615807927483311551
Technical details
Node id
n-4ce499352d70bef2 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-5cdcbb7c4ab4eee9

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:        3,         2;
  2:       12,        16,         3;
  3:       60,       120,        45,         4;
  4:      360,       960,       540,        96,        5;
  5:     2520,      8400,      6300,      1680,      175,       6;
  6:    20160,     80640,     75600,     26880,     4200,     288,      7;
  7:   181440,    846720,    952560,    423360,    88200,    9072,    441,     8;
  8:  1814400,   9676800,  12700800,   6773760,  1764000,  241920,  17640,   640,   9;
  9: 19958400, 119750400, 179625600, 111767040, 34927200, 5987520, 582120, 31680, 891, 10;
%F Formula
The row polynomials satisfy (1+n*x)*P(n,x) = (n+2+(n^2+2*n+2)*x+(n+1)*x^2)*P(n-1,x) + (x+(n+1)*x^2)*d/dx 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[((n + 2 + (n^2 + 2*n + 2)*x + (n + 1)*x^2)*P[n - 1, x] + (x + (n + 1)*x^2)*D[P[n - 1, x], x])/(1 + 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] = ((n + 2 + (n^2 + 2*n + 2)*x + (n + 1)*x^2)*v[i - 1] + (x + (n + 1)*x^2)*deriv(v[i - 1], x))/(1 + 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. [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.