Real-rooted atlas

A094262 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 + 4\,x + 3\,x^{2}\right)\,P_{n-1}(x) + \left(x + 2\,x^{2} + x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
11438
2112364015104
31281834966554201051888
4160720356091501335611200504094544032
511242523202408495021129632943432636820002569300103951254656
6125283161016406397652408616585883295722561063975579571803839220108108013513542242048
715082646347409642568052273871679210803189588288320787891388089460334326685200600400797746951891890020270251640820736

All 15 rows

Technical details
Node id
n-49a8d5132607d3b9 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-532b1e88c6581998

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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: 1,    4,      3;
  2: 1,   12,     36,      40,        15;
  3: 1,   28,    183,     496,       655,        420,        105;
  4: 1,   60,    720,    3560,      9150,      13356,      11200,        5040,          945;
  5: 1,  124,   2523,   20240,     84950,     211296,     329434,      326368,       200025,        69300,        10395;
  6: 1,  252,   8316,  101640,    639765,    2408616,    5858832,     9572256,     10639755,      7957180,      3839220,      1081080,       135135;
  7: 1,  508,  26463,  474096,   4256805,   22738716,   79210803,   189588288,    320787891,    388089460,    334326685,    200600400,     79774695,     18918900,      2027025;
  8: 1, 1020,  82440, 2110120,  26157300,  190358532,  902615280,  2965988880,   7012577430,  12192535300,  15744228456,  15088118760,  10598928340,   5306961660,   1794592800,    367567200,    34459425;
  9: 1, 2044, 253443, 9104080, 152299900, 1468699512, 9166427508, 39692623776, 124624534230, 291836790400, 518560641346, 705229331664, 734752126108, 582301119400, 345111666900, 148219431360, 43591172625, 7856748900, 654729075;
%F Formula
The row polynomials satisfy P(n,x) = (1+4*x+3*x^2)*P(n-1,x) + (x+2*x^2+x^3)*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[(1 + 4*x + 3*x^2)*P[n - 1, x] + (x + 2*x^2 + x^3)*D[P[n - 1, x], 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] = (1 + 4*x + 3*x^2)*v[i - 1] + (x + 2*x^2 + x^3)*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 - 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.