Real-rooted atlas

A132393 → D → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-256 + 256\,x - 96\,x^{2} + 16\,x^{3} - x^{4} + 256\,n - 192\,n x + 48\,n x^{2} - 4\,n x^{3} - 96\,n^{2} + 48\,n^{2} x - 6\,n^{2} x^{2} + 16\,n^{3} - 4\,n^{3} x - n^{4}\right)\,P_{n-4}(x) + \left(-175 + 148\,x - 42\,x^{2} + 4\,x^{3} + 148\,n - 84\,n x + 12\,n x^{2} - 42\,n^{2} + 12\,n^{2} x + 4\,n^{3}\right)\,P_{n-3}(x) + \left(-55 + 36\,x - 6\,x^{2} + 36\,n - 12\,n x - 6\,n^{2}\right)\,P_{n-2}(x) + \left(-10 + 4\,x + 4\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + 8\,x\)
\(P_{3}(x) = 2 + 24\,x + 27\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
2189
32242753
468816264320
5244009456401252134
6120219260755440187521615918
7720141124384847040218754536343132474
850401045443545644332162450006955296045121222032
9403208766723189348430617628061259797761872781843272912404856
103628808212608316629004631552033665625136669683241350445440328051000137607096
113628800850291203443465525382080004271162501948438805411613992928009622805500013311657600152
1239916800964350720407478535267365168645749466250288133956090468239418300057623816430192500087846172821559889520
1347900160011891543040521521079049049695283282150854500445111524001543164923335464366085463585275577000036163271347842197301265579952
1462270208001584220723207181113605121300080198144124471296950072060165777627129350147969138674176121521770371474473000121242121644716819992727444502341996704
Technical details
Node id
n-e14b11c6ee22ff96 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-8adb5a6a862fea86

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:     0;
  1:     1;
  2:     1,      8;
  3:     2,     24,      27;
  4:     6,     88,     162,      64;
  5:    24,    400,     945,     640,     125;
  6:   120,   2192,    6075,    5440,    1875,    216;
  7:   720,  14112,   43848,   47040,   21875,   4536,    343;
  8:  5040, 104544,  354564,  433216,  245000,  69552,   9604,   512;
  9: 40320, 876672, 3189348, 4306176, 2806125, 979776, 187278, 18432, 729;
%F Formula
The row polynomials satisfy P(n,x) = -(n^4-16*n^3+96*n^2-256*n+256+(4*n^3-48*n^2+192*n-256)*x+(6*n^2-48*n+96)*x^2+(4*n-16)*x^3+x^4)*P(n-4,x) + (4*n^3-42*n^2+148*n-175+(12*n^2-84*n+148)*x+(12*n-42)*x^2+4*x^3)*P(n-3,x) - (6*n^2-36*n+55+(12*n-36)*x+6*x^2)*P(n-2,x) + (4*n-10+4*x)*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 1 + 8*x, P(3,x) = 2 + 24*x + 27*x^2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[2, x_] := 1 + 8*x;
P[3, x_] := 2 + 24*x + 27*x^2;
P[n_, x_] := P[n, x] = Expand[-(n^4 - 16*n^3 + 96*n^2 - 256*n + 256 + (4*n^3 - 48*n^2 + 192*n - 256)*x + (6*n^2 - 48*n + 96)*x^2 + (4*n - 16)*x^3 + x^4)*P[n - 4, x] + (4*n^3 - 42*n^2 + 148*n - 175 + (12*n^2 - 84*n + 148)*x + (12*n - 42)*x^2 + 4*x^3)*P[n - 3, x] - (6*n^2 - 36*n + 55 + (12*n - 36)*x + 6*x^2)*P[n - 2, x] + (4*n - 10 + 4*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; v[3] = 1 + 8*x; v[4] = 2 + 24*x + 27*x^2; for(n = 4, N, my(i = n + 1); v[i] = -(n^4 - 16*n^3 + 96*n^2 - 256*n + 256 + (4*n^3 - 48*n^2 + 192*n - 256)*x + (6*n^2 - 48*n + 96)*x^2 + (4*n - 16)*x^3 + x^4)*v[i - 4] + (4*n^3 - 42*n^2 + 148*n - 175 + (12*n^2 - 84*n + 148)*x + (12*n - 42)*x^2 + 4*x^3)*v[i - 3] - (6*n^2 - 36*n + 55 + (12*n - 36)*x + 6*x^2)*v[i - 2] + (4*n - 10 + 4*x)*v[i - 1]); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.