Real-rooted atlas

delannoy → x ↦ x+1 → basis A007318

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(7 + 2\,x - 6\,n - n x + 2\,n^{2}\right)\,P_{n-1}(x) + \left(47 + 31\,x + 4\,x^{2} - 17\,n - 10\,n x - n x^{2}\right)\,P_{n-1}'(x) + \left(34 + 37\,x + 12\,x^{2} + x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1314
2117119
3393711188
413917579151409
54957774711371911900
61763331525239912112318827
7627913757126036241179930127141008
822363559435992735831130512959407311190513
97964722398527468719250584823243294535529351885076
1028366788585112238439837915089071767794170765916673914111843
1110102953468837533073948349132872663117272533529967073919182143119102600
12359821913472175227945912302690315454507725330324388155914071025711239999147188745929
131281524751962713959862631068758657997066342432529164130714682481984575150601162791177511412291516
144564217919924595539898285948546225540072377123722936310380267534082143843275515637472138192089513795511915403851

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(P_{n}(x) = \left(2 + x\right)\,P_{n-2}(x) + \left(3 + x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + x\)
Technical details
Node id
n-4f39789af2d97116 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-453c7eedd0a027bf)
Fingerprint
f6-72610dbf058ffa6b

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:     3,      1;
  2:    11,      7,      1;
  3:    39,     37,     11,      1;
  4:   139,    175,     79,     15,     1;
  5:   495,    777,    471,    137,    19,     1;
  6:  1763,   3315,   2523,    991,   211,    23,    1;
  7:  6279,  13757,  12603,   6241,  1799,   301,   27,   1;
  8: 22363,  55943,  59927,  35831, 13051,  2959,  407,  31,  1;
  9: 79647, 223985, 274687, 192505, 84823, 24329, 4535, 529, 35, 1;
%F Formula
The row polynomials satisfy n*P(n,x) = (2*n^2-6*n+7-(n-2)*x)*P(n-1,x) - (17*n-47+(10*n-31)*x+(n-4)*x^2)*d/dx P(n-1,x) + (34+37*x+12*x^2+x^3)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 - 6*n + 7 - (n - 2)*x)*P[n - 1, x] - (17*n - 47 + (10*n - 31)*x + (n - 4)*x^2)*D[P[n - 1, x], x] + (34 + 37*x + 12*x^2 + x^3)*D[P[n - 1, x], {x, 2}])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((2*n^2 - 6*n + 7 - (n - 2)*x)*v[i - 1] - (17*n - 47 + (10*n - 31)*x + (n - 4)*x^2)*deriv(v[i - 1], x) + (34 + 37*x + 12*x^2 + x^3)*deriv(deriv(v[i - 1], x), x))/(n)); 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. [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