Real-rooted atlas

A081579 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(5 - 5\,x - 7\,x^{2} - 7\,x^{3} + 2\,x^{4} + 30\,n x - 5\,n x^{2} - 17\,n x^{3} - 4\,n x^{4} + 55\,n^{2} x^{2} + 25\,n^{2} x^{3} - 5\,n^{2} x^{4} - 2\,n^{2} x^{5} + 30\,n^{3} x^{3} + 10\,n^{3} x^{4} - 2\,n^{3} x^{5} + 5\,n^{4} x^{4}\right)\,P_{n}(x) = \left(20 + 140\,x + 372\,x^{2} + 468\,x^{3} + 280\,x^{4} + 48\,x^{5} - 32\,x^{6} + 120\,n x + 660\,n x^{2} + 1332\,n x^{3} + 1128\,n x^{4} + 248\,n x^{5} - 80\,n x^{6} + 220\,n^{2} x^{2} + 900\,n^{2} x^{3} + 1140\,n^{2} x^{4} + 408\,n^{2} x^{5} - 64\,n^{2} x^{6} + 120\,n^{3} x^{3} + 360\,n^{3} x^{4} + 232\,n^{3} x^{5} - 16\,n^{3} x^{6} + 20\,n^{4} x^{4} + 40\,n^{4} x^{5}\right)\,P_{n-2}(x) + \left(10 + 30\,x + 16\,x^{2} - 60\,x^{3} - 54\,x^{4} - 2\,x^{5} - 4\,x^{6} + 60\,n x + 160\,n x^{2} + 156\,n x^{3} + 28\,n x^{4} - 8\,n x^{5} - 12\,n x^{6} + 110\,n^{2} x^{2} + 250\,n^{2} x^{3} + 180\,n^{2} x^{4} + 20\,n^{2} x^{5} - 12\,n^{2} x^{6} + 60\,n^{3} x^{3} + 100\,n^{3} x^{4} + 36\,n^{3} x^{5} - 4\,n^{3} x^{6} + 10\,n^{4} x^{4} + 10\,n^{4} x^{5}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
12810
286436108
324288504128944
4801280360025604007920
52565120201602560010400115262688
683219968100800184320134400368643136480320
72688752644642561116160120640058521611916881923577344
88704278528202982460538888742400634060822453763604482073626080512
92816010137608524800303923205518080054097920286003207864320103680051200186790400
10911363645440347443201441792003159808003931914242824908801153433602571264028672001239041318370304
112949121297612813824000065470464016831488002548482048233880371212989235204271616007966720076820482949129190379520
129543684580966453946777628717875208478412800151572971521702652313612120489984541292544014794752002364088322005401669222463390298112
13308838416059596820716093441224658124840847564800843162255361124199301129835223449656366088192208617472004855054336677117952512245761605632433230667776
1499942405596774407848345600510184652801897766912004445739417606875195801607167646105605060579328002402779136007546249216015241052160188284928012845056036864002937125683200
Technical details
Node id
n-a45cca972eade03b · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-e89b4675c1260d26

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,       8;
  2:     8,      64,      36;
  3:    24,     288,     504,      128;
  4:    80,    1280,    3600,     2560,      400;
  5:   256,    5120,   20160,    25600,    10400,     1152;
  6:   832,   19968,  100800,   184320,   134400,    36864,     3136;
  7:  2688,   75264,  464256,  1116160,  1206400,   585216,   119168,    8192;
  8:  8704,  278528, 2029824,  6053888,  8742400,  6340608,  2245376,  360448,   20736;
  9: 28160, 1013760, 8524800, 30392320, 55180800, 54097920, 28600320, 7864320, 1036800, 51200;
%F Formula
The row polynomials satisfy (5+(30*n-5)*x+(55*n^2-5*n-7)*x^2+(30*n^3+25*n^2-17*n-7)*x^3+(5*n^4+10*n^3-5*n^2-4*n+2)*x^4-(2*n^3+2*n^2)*x^5)*P(n,x) = (20+(120*n+140)*x+(220*n^2+660*n+372)*x^2+(120*n^3+900*n^2+1332*n+468)*x^3+(20*n^4+360*n^3+1140*n^2+1128*n+280)*x^4+(40*n^4+232*n^3+408*n^2+248*n+48)*x^5-(16*n^3+64*n^2+80*n+32)*x^6)*P(n-2,x) + (10+(60*n+30)*x+(110*n^2+160*n+16)*x^2+(60*n^3+250*n^2+156*n-60)*x^3+(10*n^4+100*n^3+180*n^2+28*n-54)*x^4+(10*n^4+36*n^3+20*n^2-8*n-2)*x^5-(4*n^3+12*n^2+12*n+4)*x^6)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 8*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((20 + (120*n + 140)*x + (220*n^2 + 660*n + 372)*x^2 + (120*n^3 + 900*n^2 + 1332*n + 468)*x^3 + (20*n^4 + 360*n^3 + 1140*n^2 + 1128*n + 280)*x^4 + (40*n^4 + 232*n^3 + 408*n^2 + 248*n + 48)*x^5 - (16*n^3 + 64*n^2 + 80*n + 32)*x^6)*P[n - 2, x] + (10 + (60*n + 30)*x + (110*n^2 + 160*n + 16)*x^2 + (60*n^3 + 250*n^2 + 156*n - 60)*x^3 + (10*n^4 + 100*n^3 + 180*n^2 + 28*n - 54)*x^4 + (10*n^4 + 36*n^3 + 20*n^2 - 8*n - 2)*x^5 - (4*n^3 + 12*n^2 + 12*n + 4)*x^6)*P[n - 1, x])/(5 + (30*n - 5)*x + (55*n^2 - 5*n - 7)*x^2 + (30*n^3 + 25*n^2 - 17*n - 7)*x^3 + (5*n^4 + 10*n^3 - 5*n^2 - 4*n + 2)*x^4 - (2*n^3 + 2*n^2)*x^5)]];
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 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = ((20 + (120*n + 140)*x + (220*n^2 + 660*n + 372)*x^2 + (120*n^3 + 900*n^2 + 1332*n + 468)*x^3 + (20*n^4 + 360*n^3 + 1140*n^2 + 1128*n + 280)*x^4 + (40*n^4 + 232*n^3 + 408*n^2 + 248*n + 48)*x^5 - (16*n^3 + 64*n^2 + 80*n + 32)*x^6)*v[i - 2] + (10 + (60*n + 30)*x + (110*n^2 + 160*n + 16)*x^2 + (60*n^3 + 250*n^2 + 156*n - 60)*x^3 + (10*n^4 + 100*n^3 + 180*n^2 + 28*n - 54)*x^4 + (10*n^4 + 36*n^3 + 20*n^2 - 8*n - 2)*x^5 - (4*n^3 + 12*n^2 + 12*n + 4)*x^6)*v[i - 1])/(5 + (30*n - 5)*x + (55*n^2 - 5*n - 7)*x^2 + (30*n^3 + 25*n^2 - 17*n - 7)*x^3 + (5*n^4 + 10*n^3 - 5*n^2 - 4*n + 2)*x^4 - (2*n^3 + 2*n^2)*x^5)); 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

  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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.