Real-rooted atlas

A049020 → reverse → D+1

nonnegative coefficients

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

Recurrence

\(\left(-24 + 24\,n\right)\,P_{n}(x) = \left(35 - 67\,x - 9\,x^{2} - 547\,x^{3} + 236\,x^{4} + 76\,x^{5} - 73\,n + 64\,n x + 219\,n x^{2} + 271\,n x^{3} - 217\,n x^{4} - 6\,n x^{5} + 14\,n^{2} - 27\,n^{2} x - 3\,n^{2} x^{2} + 84\,n^{2} x^{3} + 80\,n^{2} x^{4} - 16\,n^{2} x^{5} - 24\,n^{3} x^{2} - 48\,n^{3} x^{3} - 24\,n^{3} x^{4}\right)\,P_{n-2}(x) + \left(-7 + 47\,x + 46\,x^{2} - 368\,x^{3} - 237\,x^{4} + 361\,x^{5} + 38\,x^{6} + 7\,n - 42\,n x - 44\,n x^{2} + 176\,n x^{3} + 103\,n x^{4} - 102\,n x^{5} + 16\,n x^{6} + 7\,n^{2} x + 7\,n^{2} x^{2} + 32\,n^{2} x^{4} + 32\,n^{2} x^{5}\right)\,P_{n-2}'(x) + \left(7\,x^{2} - 5\,x^{3} - 69\,x^{4} + 5\,x^{5} + 62\,x^{6} - 7\,n x^{2} - 7\,n x^{3} + 24\,n x^{4} + 16\,n x^{5} - 8\,n x^{6}\right)\,P_{n-2}''(x) + \left(-31 + 130\,x - 77\,x^{2} + 38\,x^{3} + 55\,n - 124\,n x - 112\,n x^{2} + 16\,n x^{3} + 48\,n^{2} x + 48\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(7 - 12\,x - 33\,x^{2} + 86\,x^{3} - 7\,n - 32\,n x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1213
247213
372625563
411721429715337
516165555764411521963
622331171539444226189220312333
7296024487156242756624380940287782967

All 15 rows

Technical details
Node id
n-a90817ebb63d5f22 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-a2dd6bb0309a3be6

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:  2,    1;
  2:  4,    7,     2;
  3:  7,   26,    25,      5;
  4: 11,   72,   142,     97,     15;
  5: 16,  165,   555,    764,    411,      52;
  6: 22,  331,  1715,   3944,   4226,    1892,     203;
  7: 29,  602,  4487,  15624,  27566,   24380,    9402,    877;
  8: 37, 1016, 10360,  51254, 135121,  194648,  147386,  50127,   4140;
  9: 46, 1617, 21702, 145656, 539826, 1149861, 1406077, 934520, 285151, 21147;
%F Formula
The row polynomials satisfy (24*n-24)*P(n,x) = (14*n^2-73*n+35-(27*n^2-64*n+67)*x-(24*n^3+3*n^2-219*n+9)*x^2-(48*n^3-84*n^2-271*n+547)*x^3-(24*n^3-80*n^2+217*n-236)*x^4-(16*n^2+6*n-76)*x^5)*P(n-2,x) + (7*n-7+(7*n^2-42*n+47)*x+(7*n^2-44*n+46)*x^2+(176*n-368)*x^3+(32*n^2+103*n-237)*x^4+(32*n^2-102*n+361)*x^5+(16*n+38)*x^6)*d/dx P(n-2,x) - ((7*n-7)*x^2+(7*n+5)*x^3-(24*n-69)*x^4-(16*n+5)*x^5+(8*n-62)*x^6)*d^2/dx^2 P(n-2,x) + (55*n-31+(48*n^2-124*n+130)*x+(48*n^2-112*n-77)*x^2+(16*n+38)*x^3)*P(n-1,x) - (7*n-7+12*x+33*x^2+(32*n-86)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[((14*n^2 - 73*n + 35 - (27*n^2 - 64*n + 67)*x - (24*n^3 + 3*n^2 - 219*n + 9)*x^2 - (48*n^3 - 84*n^2 - 271*n + 547)*x^3 - (24*n^3 - 80*n^2 + 217*n - 236)*x^4 - (16*n^2 + 6*n - 76)*x^5)*P[n - 2, x] + (7*n - 7 + (7*n^2 - 42*n + 47)*x + (7*n^2 - 44*n + 46)*x^2 + (176*n - 368)*x^3 + (32*n^2 + 103*n - 237)*x^4 + (32*n^2 - 102*n + 361)*x^5 + (16*n + 38)*x^6)*D[P[n - 2, x], x] - ((7*n - 7)*x^2 + (7*n + 5)*x^3 - (24*n - 69)*x^4 - (16*n + 5)*x^5 + (8*n - 62)*x^6)*D[P[n - 2, x], {x, 2}] + (55*n - 31 + (48*n^2 - 124*n + 130)*x + (48*n^2 - 112*n - 77)*x^2 + (16*n + 38)*x^3)*P[n - 1, x] - (7*n - 7 + 12*x + 33*x^2 + (32*n - 86)*x^3)*D[P[n - 1, x], x])/(24*n - 24)]];
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] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = ((14*n^2 - 73*n + 35 - (27*n^2 - 64*n + 67)*x - (24*n^3 + 3*n^2 - 219*n + 9)*x^2 - (48*n^3 - 84*n^2 - 271*n + 547)*x^3 - (24*n^3 - 80*n^2 + 217*n - 236)*x^4 - (16*n^2 + 6*n - 76)*x^5)*v[i - 2] + (7*n - 7 + (7*n^2 - 42*n + 47)*x + (7*n^2 - 44*n + 46)*x^2 + (176*n - 368)*x^3 + (32*n^2 + 103*n - 237)*x^4 + (32*n^2 - 102*n + 361)*x^5 + (16*n + 38)*x^6)*deriv(v[i - 2], x) - ((7*n - 7)*x^2 + (7*n + 5)*x^3 - (24*n - 69)*x^4 - (16*n + 5)*x^5 + (8*n - 62)*x^6)*deriv(deriv(v[i - 2], x), x) + (55*n - 31 + (48*n^2 - 124*n + 130)*x + (48*n^2 - 112*n - 77)*x^2 + (16*n + 38)*x^3)*v[i - 1] - (7*n - 7 + 12*x + 33*x^2 + (32*n - 86)*x^3)*deriv(v[i - 1], x))/(24*n - 24)); 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

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  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [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
  7. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  8. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.