Real-rooted atlas

delannoy → basis A013611 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-2\,n - 2\,n x - 24\,n x^{2} - 12\,n x^{3} + 16\,n x^{4} + 4\,n^{2} + 16\,n^{2} x^{2} + 16\,n^{2} x^{3} + 8\,n^{3} x\right)\,P_{n}(x) = \left(-8 - 70\,x - 172\,x^{2} - 60\,x^{3} + 96\,x^{4} + 64\,x^{5} + 26\,n x + 116\,n x^{2} + 100\,n x^{3} + 16\,n x^{4} + 28\,n^{2} + 112\,n^{2} x + 32\,n^{2} x^{2} - 48\,n^{2} x^{3} - 8\,n^{3} - 24\,n^{3} x\right)\,P_{n-1}(x) + \left(-1 - 10\,x - 24\,x^{2} + 44\,x^{3} + 228\,x^{4} + 224\,x^{5} + 64\,x^{6} + 8\,n x + 72\,n x^{2} + 208\,n x^{3} + 208\,n x^{4} + 64\,n x^{5} + 4\,n^{2} + 32\,n^{2} x + 72\,n^{2} x^{2} + 32\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
16410
225521693
38432832064796
42611512275217282566509
5770594816256188808704102451582
621972126078160136512115456419844096399665
7612071248330368779136989184655360196608163843044408
816745227984127846438170886550016649625635307529011206553622883961
945182704564463974416808576365803524901478439763968183009284063232262144170183474
1012056121190761603611268415360181011968307791872336986112230866944920780801808793610485761254562597
1131878062361845333856026213318481827635216932864002352447488217210880012863406084524605447969177641943049180832980
128365251802776817200531295738963234485050888429199360142572994561669662310413307674624693475737621810380803481272321677721666768260773
132181114513448765407169283363309120137416453123880258252877574684672110607384576111735603200782554890243638978150410347347968150994944067108864482989129126
145655725144401660166386824011440880320523087936001677282457603876661043206535244185608036112793607127918182404448803880961867010539524843582259265095598082684354563477680725689
Technical details
Node id
n-624f11ebdb00369c · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-e241b42e92753aa4

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:     6,      4;
  2:    25,     52,      16;
  3:    84,    328,     320,       64;
  4:   261,   1512,    2752,     1728,      256;
  5:   770,   5948,   16256,    18880,     8704,     1024;
  6:  2197,  21260,   78160,   136512,   115456,    41984,     4096;
  7:  6120,  71248,  330368,   779136,   989184,   655360,   196608,    16384;
  8: 16745, 227984, 1278464,  3817088,  6550016,  6496256,  3530752,   901120,   65536;
  9: 45182, 704564, 4639744, 16808576, 36580352, 49014784, 39763968, 18300928, 4063232, 262144;
%F Formula
The row polynomials satisfy (4*n^2-2*n+(8*n^3-2*n)*x+(16*n^2-24*n)*x^2+(16*n^2-12*n)*x^3+16*n*x^4)*P(n,x) = -(8*n^3-28*n^2+8+(24*n^3-112*n^2-26*n+70)*x-(32*n^2+116*n-172)*x^2+(48*n^2-100*n+60)*x^3-(16*n+96)*x^4-64*x^5)*P(n-1,x) + (4*n^2-1+(32*n^2+8*n-10)*x+(72*n^2+72*n-24)*x^2+(32*n^2+208*n+44)*x^3+(208*n+228)*x^4+(64*n+224)*x^5+64*x^6)*d/dx 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[(-(8*n^3 - 28*n^2 + 8 + (24*n^3 - 112*n^2 - 26*n + 70)*x - (32*n^2 + 116*n - 172)*x^2 + (48*n^2 - 100*n + 60)*x^3 - (16*n + 96)*x^4 - 64*x^5)*P[n - 1, x] + (4*n^2 - 1 + (32*n^2 + 8*n - 10)*x + (72*n^2 + 72*n - 24)*x^2 + (32*n^2 + 208*n + 44)*x^3 + (208*n + 228)*x^4 + (64*n + 224)*x^5 + 64*x^6)*D[P[n - 1, x], x])/(4*n^2 - 2*n + (8*n^3 - 2*n)*x + (16*n^2 - 24*n)*x^2 + (16*n^2 - 12*n)*x^3 + 16*n*x^4)]];
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] = (-(8*n^3 - 28*n^2 + 8 + (24*n^3 - 112*n^2 - 26*n + 70)*x - (32*n^2 + 116*n - 172)*x^2 + (48*n^2 - 100*n + 60)*x^3 - (16*n + 96)*x^4 - 64*x^5)*v[i - 1] + (4*n^2 - 1 + (32*n^2 + 8*n - 10)*x + (72*n^2 + 72*n - 24)*x^2 + (32*n^2 + 208*n + 44)*x^3 + (208*n + 228)*x^4 + (64*n + 224)*x^5 + 64*x^6)*deriv(v[i - 1], x))/(4*n^2 - 2*n + (8*n^3 - 2*n)*x + (16*n^2 - 24*n)*x^2 + (16*n^2 - 12*n)*x^3 + 16*n*x^4)); 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.