Real-rooted atlas

A131689 → x ↦ x+1 → D+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(x + 2\,x^{2} + x^{3}\right)\,P_{n-2}(x) + \left(2 + 7\,x + 9\,x^{2} + 5\,x^{3} + x^{4}\right)\,P_{n-2}'(x) + \left(3 + x - x^{2}\right)\,P_{n-1}(x) + \left(2 + 3\,x + x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2213
389219
44479426171
5308765662228241987
626128311100505670144012028203
7259881009891573821246205196010440720472099
829656413624392603202266460615624005157608568050409095691
938165482024644545857702576775084378802420346480555408078624040320198113347
1054667412328972471862110690128751135012026145607241446802783592006474384079833603628804811470443
11862440068580591746917285512262299067821403320388492024617499480122684738404018896000814060800889056003628800128876001379
121485710008411064591139936895337536272595931700693454313040082410545712050591932776021491521128061312204800109970784001077753600399168003773325784011
13277474957988226519198132583639929223421844767316158827026397074424276493362393602019726647136010571699033280391147251264098804059200015898861440014130547200479001600119872143109507
145584100659412495897905166312008721607465304909310246322308066998736596809396805494476407966460646144004962968327116802262908699251207413600370176016798945363200245085160320019926466560062270208004106182559267883
15120462266974148115620127726194950980198844975421367645997392766024914821165186680325653440893981203144563631144936022741730339328000123436546257024004983877567468800146466794073600030087407750400040145603097600300765104640087178291200150854989951869859
Technical details
Node id
n-d57417214bfca832 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-c8df2948d24e794c

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:        2,         1;
   3:        8,         9,         2;
   4:       44,        79,        42,          6;
   5:      308,       765,       662,        228,         24;
   6:     2612,      8311,     10050,       5670,       1440,       120;
   7:    25988,    100989,    157382,     124620,      51960,     10440,       720;
   8:   296564,   1362439,   2603202,    2664606,    1562400,    515760,     85680,     5040;
   9:  3816548,  20246445,  45857702,   57677508,   43788024,  20346480,   5554080,   786240,   40320;
  10: 54667412, 328972471, 862110690, 1287511350, 1202614560, 724144680, 278359200, 64743840, 7983360, 362880;
%F Formula
The row polynomials satisfy P(n,x) = (x+2*x^2+x^3)*P(n-2,x) + (2+7*x+9*x^2+5*x^3+x^4)*d/dx P(n-2,x) + (3+x-x^2)*P(n-1,x) + (2+3*x+x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[(x + 2*x^2 + x^3)*P[n - 2, x] + (2 + 7*x + 9*x^2 + 5*x^3 + x^4)*D[P[n - 2, x], x] + (3 + x - x^2)*P[n - 1, x] + (2 + 3*x + x^2)*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; v[2] = 2 + x; for(n = 3, N, my(i = n); v[i] = (x + 2*x^2 + x^3)*v[i - 2] + (2 + 7*x + 9*x^2 + 5*x^3 + x^4)*deriv(v[i - 2], x) + (3 + x - x^2)*v[i - 1] + (2 + 3*x + x^2)*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

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. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.