Real-rooted atlas

A081579 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(n + 3\,n x + 3\,n x^{2} + n x^{3}\right)\,P_{n}(x) = \left(46 + 326\,x + 940\,x^{2} + 1420\,x^{3} + 1190\,x^{4} + 526\,x^{5} + 96\,x^{6} - 64\,n - 456\,n x - 1330\,n x^{2} - 2038\,n x^{3} - 1734\,n x^{4} - 778\,n x^{5} - 144\,n x^{6} + 20\,n^{2} + 140\,n^{2} x + 408\,n^{2} x^{2} + 632\,n^{2} x^{3} + 548\,n^{2} x^{4} + 252\,n^{2} x^{5} + 48\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(45 + 423\,x + 1632\,x^{2} + 3390\,x^{3} + 4125\,x^{4} + 2955\,x^{5} + 1158\,x^{6} + 192\,x^{7} - 20\,n - 190\,n x - 744\,n x^{2} - 1572\,n x^{3} - 1948\,n x^{4} - 1422\,n x^{5} - 568\,n x^{6} - 96\,n x^{7}\right)\,P_{n-1}'(x) + \left(5 + 60\,x + 301\,x^{2} + 828\,x^{3} + 1375\,x^{4} + 1420\,x^{5} + 895\,x^{6} + 316\,x^{7} + 48\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1246
28322868
324144272160600
480640184022409765776
5256256099201856016736582453856
683299844864012288016953612108835008507968
7268837632220928704000131353614341128483842099204771200
887041392649566723681280867097612793856115450885826560125977644882176
928160506880398848017991680512363529547264011636736089456640393651207558144421971456
1091136182272016163840836403202795038726299770889695590401005936640673361920262651904453498883968058368
11294912648806464020480373800960143424307237955420167066894336925548544083556454404952305664173461913627209728037311436800
129543682290483224890572816187392007013916672213237104644662787276873874964480841563750406722283110435742633984113600921601632587776350846488576
133088384802979849528606726830489600329888235521133187563522847577538565299001753607301256888327356365373445268197048322539606835207387453030497955184643299044909056
14999424027983872036004659202820472832015024196812857553518592016346483916803496261713920565801148416068936687288326224597893120403919667200017807952281604775316684805877312716831021357088768
Technical details
Node id
n-c3db691544f4943a · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-da2b4652998f9682

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,      4;
  2:     8,     32,      28;
  3:    24,    144,     272,      160;
  4:    80,    640,    1840,     2240,      976;
  5:   256,   2560,    9920,    18560,    16736,     5824;
  6:   832,   9984,   48640,   122880,   169536,   121088,     35008;
  7:  2688,  37632,  220928,   704000,  1313536,  1434112,    848384,   209920;
  8:  8704, 139264,  956672,  3681280,  8670976, 12793856,  11545088,  5826560,  1259776;
  9: 28160, 506880, 3988480, 17991680, 51236352, 95472640, 116367360, 89456640, 39365120, 7558144;
%F Formula
The row polynomials satisfy (n+3*n*x+3*n*x^2+n*x^3)*P(n,x) = (20*n^2-64*n+46+(140*n^2-456*n+326)*x+(408*n^2-1330*n+940)*x^2+(632*n^2-2038*n+1420)*x^3+(548*n^2-1734*n+1190)*x^4+(252*n^2-778*n+526)*x^5+(48*n^2-144*n+96)*x^6)*P(n-1,x) - (20*n-45+(190*n-423)*x+(744*n-1632)*x^2+(1572*n-3390)*x^3+(1948*n-4125)*x^4+(1422*n-2955)*x^5+(568*n-1158)*x^6+(96*n-192)*x^7)*d/dx P(n-1,x) + (5+60*x+301*x^2+828*x^3+1375*x^4+1420*x^5+895*x^6+316*x^7+48*x^8)*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[((20*n^2 - 64*n + 46 + (140*n^2 - 456*n + 326)*x + (408*n^2 - 1330*n + 940)*x^2 + (632*n^2 - 2038*n + 1420)*x^3 + (548*n^2 - 1734*n + 1190)*x^4 + (252*n^2 - 778*n + 526)*x^5 + (48*n^2 - 144*n + 96)*x^6)*P[n - 1, x] - (20*n - 45 + (190*n - 423)*x + (744*n - 1632)*x^2 + (1572*n - 3390)*x^3 + (1948*n - 4125)*x^4 + (1422*n - 2955)*x^5 + (568*n - 1158)*x^6 + (96*n - 192)*x^7)*D[P[n - 1, x], x] + (5 + 60*x + 301*x^2 + 828*x^3 + 1375*x^4 + 1420*x^5 + 895*x^6 + 316*x^7 + 48*x^8)*D[P[n - 1, x], {x, 2}])/(n + 3*n*x + 3*n*x^2 + n*x^3)]];
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] = ((20*n^2 - 64*n + 46 + (140*n^2 - 456*n + 326)*x + (408*n^2 - 1330*n + 940)*x^2 + (632*n^2 - 2038*n + 1420)*x^3 + (548*n^2 - 1734*n + 1190)*x^4 + (252*n^2 - 778*n + 526)*x^5 + (48*n^2 - 144*n + 96)*x^6)*v[i - 1] - (20*n - 45 + (190*n - 423)*x + (744*n - 1632)*x^2 + (1572*n - 3390)*x^3 + (1948*n - 4125)*x^4 + (1422*n - 2955)*x^5 + (568*n - 1158)*x^6 + (96*n - 192)*x^7)*deriv(v[i - 1], x) + (5 + 60*x + 301*x^2 + 828*x^3 + 1375*x^4 + 1420*x^5 + 895*x^6 + 316*x^7 + 48*x^8)*deriv(deriv(v[i - 1], x), x))/(n + 3*n*x + 3*n*x^2 + n*x^3)); 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. [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