Real-rooted atlas

Veronese parent of A109447 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(314\,n + 116\,n x + 170\,n^{2} + 260\,n^{2} x + 108\,n^{3} x\right)\,P_{n}(x) = \left(9\,n + 900\,n x + 1079\,n x^{2} + 92\,n x^{3} - 228\,n x^{4} - 332\,n^{2} + 34\,n^{2} x + 1287\,n^{2} x^{2} + 200\,n^{2} x^{3} - 405\,n^{2} x^{4} - 197\,n^{3} - 454\,n^{3} x - 167\,n^{3} x^{2} + 56\,n^{3} x^{3} - 191\,n^{3} x^{4} - 128\,n^{4} x - 275\,n^{4} x^{2} - 100\,n^{4} x^{3} - 7\,n^{4} x^{4} - 44\,n^{5} x^{2} - 56\,n^{5} x^{3} + 7\,n^{5} x^{4} - 8\,n^{6} x^{3}\right)\,P_{n-1}(x) + \left(74\,n + 508\,n x + 478\,n x^{2} + 448\,n x^{3} + 860\,n x^{4} + 456\,n x^{5} + 56\,n^{2} + 356\,n^{2} x + 936\,n^{2} x^{2} + 1014\,n^{2} x^{3} + 732\,n^{2} x^{4} + 354\,n^{2} x^{5} + 52\,n^{3} x + 370\,n^{3} x^{2} + 558\,n^{3} x^{3} + 268\,n^{3} x^{4} + 28\,n^{3} x^{5} + 56\,n^{4} x^{2} + 152\,n^{4} x^{3} + 82\,n^{4} x^{4} - 14\,n^{4} x^{5} + 16\,n^{5} x^{3} + 16\,n^{5} x^{4}\right)\,P_{n-1}'(x) + \left(1 - 36\,x - 382\,x^{2} - 1148\,x^{3} - 1527\,x^{4} - 952\,x^{5} - 228\,x^{6} - 3\,n + 32\,n x - 261\,n x^{2} - 1116\,n x^{3} - 1483\,n x^{4} - 840\,n x^{5} - 177\,n x^{6} + 28\,n^{2} x - 31\,n^{2} x^{2} - 334\,n^{2} x^{3} - 477\,n^{2} x^{4} - 216\,n^{2} x^{5} - 14\,n^{2} x^{6} - 12\,n^{3} x^{2} - 72\,n^{3} x^{3} - 101\,n^{3} x^{4} - 34\,n^{3} x^{5} + 7\,n^{3} x^{6} - 8\,n^{4} x^{3} - 16\,n^{4} x^{4} - 8\,n^{4} x^{5}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 4\,x + 9\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8Pn(1)
004913
118918
204274825104
3116546425160
4044516025018049688
5124135320375216491120
6046333687512601029448814096
7132252896175020161372512816912

All 15 rows

Technical details
Node id
n-aec90f69622c791f · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-99b0e02b99873e6a

Observed on the stored rows ?

  • Degrees: 2, 2, 4, 4, 6, 6, 8, 8, 10, 10, 12, 12, 14, 14, 16 (irregular)
  • 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,  4,   9;
  1: 1,  8,   9;
  2: 0,  4,  27,   48,   25;
  3: 1, 16,  54,   64,   25;
  4: 0,  4,  45,  160,  250,  180,    49;
  5: 1, 24, 135,  320,  375,  216,    49;
  6: 0,  4,  63,  336,  875, 1260,  1029,  448,   81;
  7: 1, 32, 252,  896, 1750, 2016,  1372,  512,   81;
  8: 0,  4,  81,  576, 2100, 4536,  6174, 5376, 2916,  900, 121;
  9: 1, 40, 405, 1920, 5250, 9072, 10290, 7680, 3645, 1000, 121;
%F Formula
The row polynomials satisfy (170*n^2+314*n+(108*n^3+260*n^2+116*n)*x)*P(n,x) = -(197*n^3+332*n^2-9*n+(128*n^4+454*n^3-34*n^2-900*n)*x+(44*n^5+275*n^4+167*n^3-1287*n^2-1079*n)*x^2+(8*n^6+56*n^5+100*n^4-56*n^3-200*n^2-92*n)*x^3-(7*n^5-7*n^4-191*n^3-405*n^2-228*n)*x^4)*P(n-1,x) + (56*n^2+74*n+(52*n^3+356*n^2+508*n)*x+(56*n^4+370*n^3+936*n^2+478*n)*x^2+(16*n^5+152*n^4+558*n^3+1014*n^2+448*n)*x^3+(16*n^5+82*n^4+268*n^3+732*n^2+860*n)*x^4-(14*n^4-28*n^3-354*n^2-456*n)*x^5)*d/dx P(n-1,x) - (3*n-1-(28*n^2+32*n-36)*x+(12*n^3+31*n^2+261*n+382)*x^2+(8*n^4+72*n^3+334*n^2+1116*n+1148)*x^3+(16*n^4+101*n^3+477*n^2+1483*n+1527)*x^4+(8*n^4+34*n^3+216*n^2+840*n+952)*x^5-(7*n^3-14*n^2-177*n-228)*x^6)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 4*x + 9*x^2.
%t Mathematica
P[0, x_] := 4*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(197*n^3 + 332*n^2 - 9*n + (128*n^4 + 454*n^3 - 34*n^2 - 900*n)*x + (44*n^5 + 275*n^4 + 167*n^3 - 1287*n^2 - 1079*n)*x^2 + (8*n^6 + 56*n^5 + 100*n^4 - 56*n^3 - 200*n^2 - 92*n)*x^3 - (7*n^5 - 7*n^4 - 191*n^3 - 405*n^2 - 228*n)*x^4)*P[n - 1, x] + (56*n^2 + 74*n + (52*n^3 + 356*n^2 + 508*n)*x + (56*n^4 + 370*n^3 + 936*n^2 + 478*n)*x^2 + (16*n^5 + 152*n^4 + 558*n^3 + 1014*n^2 + 448*n)*x^3 + (16*n^5 + 82*n^4 + 268*n^3 + 732*n^2 + 860*n)*x^4 - (14*n^4 - 28*n^3 - 354*n^2 - 456*n)*x^5)*D[P[n - 1, x], x] - (3*n - 1 - (28*n^2 + 32*n - 36)*x + (12*n^3 + 31*n^2 + 261*n + 382)*x^2 + (8*n^4 + 72*n^3 + 334*n^2 + 1116*n + 1148)*x^3 + (16*n^4 + 101*n^3 + 477*n^2 + 1483*n + 1527)*x^4 + (8*n^4 + 34*n^3 + 216*n^2 + 840*n + 952)*x^5 - (7*n^3 - 14*n^2 - 177*n - 228)*x^6)*D[P[n - 1, x], {x, 2}])/(170*n^2 + 314*n + (108*n^3 + 260*n^2 + 116*n)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 4*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = (-(197*n^3 + 332*n^2 - 9*n + (128*n^4 + 454*n^3 - 34*n^2 - 900*n)*x + (44*n^5 + 275*n^4 + 167*n^3 - 1287*n^2 - 1079*n)*x^2 + (8*n^6 + 56*n^5 + 100*n^4 - 56*n^3 - 200*n^2 - 92*n)*x^3 - (7*n^5 - 7*n^4 - 191*n^3 - 405*n^2 - 228*n)*x^4)*v[i - 1] + (56*n^2 + 74*n + (52*n^3 + 356*n^2 + 508*n)*x + (56*n^4 + 370*n^3 + 936*n^2 + 478*n)*x^2 + (16*n^5 + 152*n^4 + 558*n^3 + 1014*n^2 + 448*n)*x^3 + (16*n^5 + 82*n^4 + 268*n^3 + 732*n^2 + 860*n)*x^4 - (14*n^4 - 28*n^3 - 354*n^2 - 456*n)*x^5)*deriv(v[i - 1], x) - (3*n - 1 - (28*n^2 + 32*n - 36)*x + (12*n^3 + 31*n^2 + 261*n + 382)*x^2 + (8*n^4 + 72*n^3 + 334*n^2 + 1116*n + 1148)*x^3 + (16*n^4 + 101*n^3 + 477*n^2 + 1483*n + 1527)*x^4 + (8*n^4 + 34*n^3 + 216*n^2 + 840*n + 952)*x^5 - (7*n^3 - 14*n^2 - 177*n - 228)*x^6)*deriv(deriv(v[i - 1], x), x))/(170*n^2 + 314*n + (108*n^3 + 260*n^2 + 116*n)*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.