Real-rooted atlas

A028421 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-81 + 108\,x - 54\,x^{2} + 12\,x^{3} - x^{4} + 108\,n - 108\,n x + 36\,n x^{2} - 4\,n x^{3} - 54\,n^{2} + 36\,n^{2} x - 6\,n^{2} x^{2} + 12\,n^{3} - 4\,n^{3} x - n^{4}\right)\,P_{n-4}(x) + \left(-65 + 76\,x - 30\,x^{2} + 4\,x^{3} + 76\,n - 60\,n x + 12\,n x^{2} - 30\,n^{2} + 12\,n^{2} x + 4\,n^{3}\right)\,P_{n-3}(x) + \left(-25 + 24\,x - 6\,x^{2} + 24\,n - 12\,n x - 6\,n^{2}\right)\,P_{n-2}(x) + \left(-6 + 4\,x + 4\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 8\,x\)
\(P_{2}(x) = 2 + 24\,x + 27\,x^{2}\)
\(P_{3}(x) = 6 + 88\,x + 162\,x^{2} + 64\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1189
22242753
368816264320
4244009456401252134
5120219260755440187521615918
6720141124384847040218754536343132474
750401045443545644332162450006955296045121222032
8403208766723189348430617628061259797761872781843272912404856
93628808212608316629004631552033665625136669683241350445440328051000137607096
103628800850291203443465525382080004271162501948438805411613992928009622805500013311657600152
1139916800964350720407478535267365168645749466250288133956090468239418300057623816430192500087846172821559889520
1247900160011891543040521521079049049695283282150854500445111524001543164923335464366085463585275577000036163271347842197301265579952
1362270208001584220723207181113605121300080198144124471296950072060165777627129350147969138674176121521770371474473000121242121644716819992727444502341996704
14871782912002267725178880105882335412481990331266560019965200710000122392872201604942397764304137290433280026857206733537312275000365991325724766560010995985288120337571676042912864
Technical details
Node id
n-08fcb9967597bac2 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-8adb5a6a862fea86

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:      1,       8;
  2:      2,      24,       27;
  3:      6,      88,      162,       64;
  4:     24,     400,      945,      640,      125;
  5:    120,    2192,     6075,     5440,     1875,      216;
  6:    720,   14112,    43848,    47040,    21875,     4536,     343;
  7:   5040,  104544,   354564,   433216,   245000,    69552,    9604,    512;
  8:  40320,  876672,  3189348,  4306176,  2806125,   979776,  187278,  18432,   729;
  9: 362880, 8212608, 31662900, 46315520, 33665625, 13666968, 3241350, 445440, 32805, 1000;
%F Formula
The row polynomials satisfy P(n,x) = -(n^4-12*n^3+54*n^2-108*n+81+(4*n^3-36*n^2+108*n-108)*x+(6*n^2-36*n+54)*x^2+(4*n-12)*x^3+x^4)*P(n-4,x) + (4*n^3-30*n^2+76*n-65+(12*n^2-60*n+76)*x+(12*n-30)*x^2+4*x^3)*P(n-3,x) - (6*n^2-24*n+25+(12*n-24)*x+6*x^2)*P(n-2,x) + (4*n-6+4*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + 8*x, P(2,x) = 2 + 24*x + 27*x^2, P(3,x) = 6 + 88*x + 162*x^2 + 64*x^3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 8*x;
P[2, x_] := 2 + 24*x + 27*x^2;
P[3, x_] := 6 + 88*x + 162*x^2 + 64*x^3;
P[n_, x_] := P[n, x] = Expand[-(n^4 - 12*n^3 + 54*n^2 - 108*n + 81 + (4*n^3 - 36*n^2 + 108*n - 108)*x + (6*n^2 - 36*n + 54)*x^2 + (4*n - 12)*x^3 + x^4)*P[n - 4, x] + (4*n^3 - 30*n^2 + 76*n - 65 + (12*n^2 - 60*n + 76)*x + (12*n - 30)*x^2 + 4*x^3)*P[n - 3, x] - (6*n^2 - 24*n + 25 + (12*n - 24)*x + 6*x^2)*P[n - 2, x] + (4*n - 6 + 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] = 1; v[2] = 1 + 8*x; v[3] = 2 + 24*x + 27*x^2; v[4] = 6 + 88*x + 162*x^2 + 64*x^3; for(n = 4, N, my(i = n + 1); v[i] = -(n^4 - 12*n^3 + 54*n^2 - 108*n + 81 + (4*n^3 - 36*n^2 + 108*n - 108)*x + (6*n^2 - 36*n + 54)*x^2 + (4*n - 12)*x^3 + x^4)*v[i - 4] + (4*n^3 - 30*n^2 + 76*n - 65 + (12*n^2 - 60*n + 76)*x + (12*n - 30)*x^2 + 4*x^3)*v[i - 3] - (6*n^2 - 24*n + 25 + (12*n - 24)*x + 6*x^2)*v[i - 2] + (4*n - 6 + 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. [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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.