Real-rooted atlas

delannoy → basis A013609 → D

nonnegative coefficients

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

Recurrence

\(\left(-1 + x^{2} + n - n x - n x^{2} + n^{2} x\right)\,P_{n}(x) = \left(-5 - 16\,x - 15\,x^{2} - 4\,x^{3} + 9\,n + 21\,n x + 9\,n x^{2} - 2\,n^{2} - 3\,n^{2} x\right)\,P_{n-1}(x) + \left(-2 - 10\,x - 17\,x^{2} - 11\,x^{3} - 2\,x^{4} + 2\,n + 8\,n x + 9\,n x^{2} + 2\,n x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
122
210818
3366424124
411632026464764
5350131217048961604422
61014480885687104272038424598
72856163843720042496256007680896133112
878805305614654421414417792083712206082048705912
92140216544053860896076810230406635522553605324846083686026
1057410501064187924839627525161280429312022749447393281336321024019013018
111524601482816629661615333888236544002409216016370816731545620551683276802252897103988
12401532430617620424120564328321007116801216350721012488965798912022367232552960078848049152491883892
1310501661231113664511832199454848404494560565840896558898816392138752193724928656588801448550418677761064962474544590
142730350347352401992997206818384001549420000246581184028231571202351175680142260480061705216018637414437109760436633622937612375904926
Technical details
Node id
n-77f99a0e6e995bca · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-fcfcffd993ba3756

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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;
  1:     2;
  2:    10,      8;
  3:    36,     64,     24;
  4:   116,    320,    264,     64;
  5:   350,   1312,   1704,    896,     160;
  6:  1014,   4808,   8568,   7104,    2720,    384;
  7:  2856,  16384,  37200,  42496,   25600,   7680,    896;
  8:  7880,  53056, 146544, 214144,  177920,  83712,  20608,  2048;
  9: 21402, 165440, 538608, 960768, 1023040, 663552, 255360, 53248, 4608;
%F Formula
The row polynomials satisfy (n-1+(n^2-n)*x-(n-1)*x^2)*P(n,x) = -(2*n^2-9*n+5+(3*n^2-21*n+16)*x-(9*n-15)*x^2+4*x^3)*P(n-1,x) + (2*n-2+(8*n-10)*x+(9*n-17)*x^2+(2*n-11)*x^3-2*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 - 9*n + 5 + (3*n^2 - 21*n + 16)*x - (9*n - 15)*x^2 + 4*x^3)*P[n - 1, x] + (2*n - 2 + (8*n - 10)*x + (9*n - 17)*x^2 + (2*n - 11)*x^3 - 2*x^4)*D[P[n - 1, x], x])/(n - 1 + (n^2 - n)*x - (n - 1)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n^2 - 9*n + 5 + (3*n^2 - 21*n + 16)*x - (9*n - 15)*x^2 + 4*x^3)*v[i - 1] + (2*n - 2 + (8*n - 10)*x + (9*n - 17)*x^2 + (2*n - 11)*x^3 - 2*x^4)*deriv(v[i - 1], x))/(n - 1 + (n^2 - n)*x - (n - 1)*x^2)); 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
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.