Real-rooted atlas

A132393 → D+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-256 + 256\,x - 96\,x^{2} + 16\,x^{3} - x^{4} + 256\,n - 192\,n x + 48\,n x^{2} - 4\,n x^{3} - 96\,n^{2} + 48\,n^{2} x - 6\,n^{2} x^{2} + 16\,n^{3} - 4\,n^{3} x - n^{4}\right)\,P_{n-4}(x) + \left(-175 + 148\,x - 42\,x^{2} + 4\,x^{3} + 148\,n - 84\,n x + 12\,n x^{2} - 42\,n^{2} + 12\,n^{2} x + 4\,n^{3}\right)\,P_{n-3}(x) + \left(-55 + 36\,x - 6\,x^{2} + 36\,n - 12\,n x - 6\,n^{2}\right)\,P_{n-2}(x) + \left(-10 + 4\,x + 4\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 4\,x\)
\(P_{2}(x) = 1 + 12\,x + 9\,x^{2}\)
\(P_{3}(x) = 2 + 32\,x + 54\,x^{2} + 16\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1145
2112922
32325416104
4611226116025564
52449613951200375363526
612026728541904040007564925178
77201699259724730244025010836137264202982
85040124704472176643328414225140112253822304811827352
9403201037952417560461961604488225178794040954253376364510018192864
10362880966412840902084650787205175762523362668634172710502401032755500121198628968
113628800995443204400043127422652166373537503178533609831683419390272243243018700079861442360984424
12399168001124017920515967991291512044808380918150453718584015583196093518042885276769351920003207711123216930361338864
1347900160013807549440655300938241214019056641175012267006817059849625529086583642581139211074158811307163001036449752603215379196420094311984
14622702080018333015552089633619187217256276710401752556796900107907899299243476332199311975903596823090131350314143830029965335419564272830466205802256224230825632
Technical details
Node id
n-677cdc00c72f0793 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-47c2124912aa21a8

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,       4;
  2:     1,      12,       9;
  3:     2,      32,      54,      16;
  4:     6,     112,     261,     160,      25;
  5:    24,     496,    1395,    1200,     375,      36;
  6:   120,    2672,    8541,    9040,    4000,     756,     49;
  7:   720,   16992,   59724,   73024,   40250,   10836,   1372,    64;
  8:  5040,  124704,  472176,  643328,  414225,  140112,  25382,  2304,   81;
  9: 40320, 1037952, 4175604, 6196160, 4488225, 1787940, 409542, 53376, 3645, 100;
%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) = 1, P(1,x) = 1 + 4*x, P(2,x) = 1 + 12*x + 9*x^2, P(3,x) = 2 + 32*x + 54*x^2 + 16*x^3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[2, x_] := 1 + 12*x + 9*x^2;
P[3, x_] := 2 + 32*x + 54*x^2 + 16*x^3;
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] = 1; v[2] = 1 + 4*x; v[3] = 1 + 12*x + 9*x^2; v[4] = 2 + 32*x + 54*x^2 + 16*x^3; 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

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.