Real-rooted atlas

A026729 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(25 - 170\,x + 136\,x^{2} + 208\,x^{3} + 16\,x^{4} - 32\,x^{5} + 150\,n x - 440\,n x^{2} - 144\,n x^{3} + 224\,n x^{4} + 32\,n x^{5} + 240\,n^{2} x^{2} - 200\,n^{2} x^{3} - 208\,n^{2} x^{4} + 32\,n^{2} x^{5} + 120\,n^{3} x^{3} - 32\,n^{3} x^{5} + 16\,n^{4} x^{4}\right)\,P_{n}(x) = \left(25 + 30\,x - 104\,x^{2} - 144\,x^{3} + 16\,x^{4} + 96\,x^{5} + 150\,n x + 340\,n x^{2} - 104\,n x^{3} - 496\,n x^{4} - 256\,n x^{5} + 240\,n^{2} x^{2} + 640\,n^{2} x^{3} + 208\,n^{2} x^{4} - 288\,n^{2} x^{5} - 128\,n^{2} x^{6} + 120\,n^{3} x^{3} + 304\,n^{3} x^{4} + 96\,n^{3} x^{5} - 64\,n^{3} x^{6} + 16\,n^{4} x^{4} + 32\,n^{4} x^{5}\right)\,P_{n-2}(x) + \left(25 + 30\,x - 104\,x^{2} - 144\,x^{3} + 16\,x^{4} + 96\,x^{5} + 150\,n x + 200\,n x^{2} - 544\,n x^{3} - 768\,n x^{4} - 160\,n x^{5} + 240\,n^{2} x^{2} + 520\,n^{2} x^{3} - 96\,n^{2} x^{4} - 384\,n^{2} x^{5} - 64\,n^{2} x^{6} + 120\,n^{3} x^{3} + 272\,n^{3} x^{4} + 32\,n^{3} x^{5} - 64\,n^{3} x^{6} + 16\,n^{4} x^{4} + 32\,n^{4} x^{5}\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2189
32243662
4356180128367
551205768964001997
68240154837123600115210260
7134643744121601840012672313650589
8218728460346887160077184407688192241785
93416081821690240236400349056288512122880207361127682
105529203783621977669840013132801489600991232352512512005156811
11895240764285093121903600435456063033605758976319334497280012390423201613
121449312150984113536048804001316160023222080271073282063232097792002601984294912102975624
1323316416292896245324811923600370540807720832010999398410701849669580800287457286782976692224451763001
143772874455965651668482801840098611200237128640399450112474688512394547200223398912816906241730560016056321962200457
15610500401055772106513926375480025069824068364800013305692161871237376190044160013753344006886195202256650244335206436864008448764454
Technical details
Node id
n-918d8c625e61ba01 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-8f4ac798ea66d40f

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:  1;
   2:  1,    8;
   3:  2,   24,    36;
   4:  3,   56,   180,    128;
   5:  5,  120,   576,    896,    400;
   6:  8,  240,  1548,   3712,   3600,    1152;
   7: 13,  464,  3744,  12160,  18400,   12672,    3136;
   8: 21,  872,  8460,  34688,  71600,   77184,   40768,   8192;
   9: 34, 1608, 18216,  90240, 236400,  349056,  288512, 122880,  20736;
  10: 55, 2920, 37836, 219776, 698400, 1313280, 1489600, 991232, 352512, 51200;
%F Formula
The row polynomials satisfy (25+(150*n-170)*x+(240*n^2-440*n+136)*x^2+(120*n^3-200*n^2-144*n+208)*x^3+(16*n^4-208*n^2+224*n+16)*x^4-(32*n^3-32*n^2-32*n+32)*x^5)*P(n,x) = (25+(150*n+30)*x+(240*n^2+340*n-104)*x^2+(120*n^3+640*n^2-104*n-144)*x^3+(16*n^4+304*n^3+208*n^2-496*n+16)*x^4+(32*n^4+96*n^3-288*n^2-256*n+96)*x^5-(64*n^3+128*n^2)*x^6)*P(n-2,x) + (25+(150*n+30)*x+(240*n^2+200*n-104)*x^2+(120*n^3+520*n^2-544*n-144)*x^3+(16*n^4+272*n^3-96*n^2-768*n+16)*x^4+(32*n^4+32*n^3-384*n^2-160*n+96)*x^5-(64*n^3+64*n^2)*x^6)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 8*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((25 + (150*n + 30)*x + (240*n^2 + 340*n - 104)*x^2 + (120*n^3 + 640*n^2 - 104*n - 144)*x^3 + (16*n^4 + 304*n^3 + 208*n^2 - 496*n + 16)*x^4 + (32*n^4 + 96*n^3 - 288*n^2 - 256*n + 96)*x^5 - (64*n^3 + 128*n^2)*x^6)*P[n - 2, x] + (25 + (150*n + 30)*x + (240*n^2 + 200*n - 104)*x^2 + (120*n^3 + 520*n^2 - 544*n - 144)*x^3 + (16*n^4 + 272*n^3 - 96*n^2 - 768*n + 16)*x^4 + (32*n^4 + 32*n^3 - 384*n^2 - 160*n + 96)*x^5 - (64*n^3 + 64*n^2)*x^6)*P[n - 1, x])/(25 + (150*n - 170)*x + (240*n^2 - 440*n + 136)*x^2 + (120*n^3 - 200*n^2 - 144*n + 208)*x^3 + (16*n^4 - 208*n^2 + 224*n + 16)*x^4 - (32*n^3 - 32*n^2 - 32*n + 32)*x^5)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 8*x; for(n = 3, N, my(i = n); v[i] = ((25 + (150*n + 30)*x + (240*n^2 + 340*n - 104)*x^2 + (120*n^3 + 640*n^2 - 104*n - 144)*x^3 + (16*n^4 + 304*n^3 + 208*n^2 - 496*n + 16)*x^4 + (32*n^4 + 96*n^3 - 288*n^2 - 256*n + 96)*x^5 - (64*n^3 + 128*n^2)*x^6)*v[i - 2] + (25 + (150*n + 30)*x + (240*n^2 + 200*n - 104)*x^2 + (120*n^3 + 520*n^2 - 544*n - 144)*x^3 + (16*n^4 + 272*n^3 - 96*n^2 - 768*n + 16)*x^4 + (32*n^4 + 32*n^3 - 384*n^2 - 160*n + 96)*x^5 - (64*n^3 + 64*n^2)*x^6)*v[i - 1])/(25 + (150*n - 170)*x + (240*n^2 - 440*n + 136)*x^2 + (120*n^3 - 200*n^2 - 144*n + 208)*x^3 + (16*n^4 - 208*n^2 + 224*n + 16)*x^4 - (32*n^3 - 32*n^2 - 32*n + 32)*x^5)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.