Real-rooted atlas

A207823 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(\left(20\,n - 32\,n x - 16\,n x^{2} + 16\,n^{3}\right)\,P_{n}(x) = \left(6 - 124\,x - 88\,x^{2} - 16\,x^{3} + 86\,n - 92\,n x - 88\,n x^{2} - 16\,n x^{3} + 120\,n^{2} + 48\,n^{2} x + 40\,n^{3} + 16\,n^{3} x\right)\,P_{n-1}(x) + \left(-231 - 556\,x - 448\,x^{2} - 144\,x^{3} - 16\,x^{4} + 84\,n + 80\,n x + 16\,n x^{2} + 84\,n^{2} + 80\,n^{2} x + 16\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1729
24428476
3261266848619
4149121161068224164915
58296151761066435685603238296
645269101668921244288010720134464294069
72433476489427223404334801506403004831361282232061
81292676399522052836603895360174464048339280192716825616782564
96800785239171103667932032135440176797606310752145331220633616128512125199455
103549379914001092024445476024842416016242184071202432211160324159488516096358401024927836391
11184000208804780688157679944018248036801384012640720966848262528896665710081145472012620807884820486837261104
1294838468145560159129901823016128628196801111124392067144631042904983424904258560200229120305817603030016172032409650138009321
1348639783992546278239460805006296876406031528499234056058499435232293585356801085650944029502950405797017607959142471639043727368192366096324209
1424837905884140747411084366380776532580376211776624595465984482671570240275902347840118338401280381980121609209277440162619494420276838416719872802816163842663103882620
Technical details
Node id
n-21c532627837f64d · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-593dc84c7ba4d89a

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:       7,        2;
  2:      44,       28,        4;
  3:     261,      266,       84,        8;
  4:    1491,     2116,     1068,      224,       16;
  5:    8296,    15176,    10664,     3568,      560,      32;
  6:   45269,   101668,    92124,    42880,    10720,    1344,      64;
  7:  243347,   648942,   722340,   433480,   150640,   30048,    3136,    128;
  8: 1292676,  3995220,  5283660,  3895360,  1744640,  483392,   80192,   7168,   256;
  9: 6800785, 23917110, 36679320, 32135440, 17679760, 6310752, 1453312, 206336, 16128, 512;
%F Formula
The row polynomials satisfy (16*n^3+20*n-32*n*x-16*n*x^2)*P(n,x) = (40*n^3+120*n^2+86*n+6+(16*n^3+48*n^2-92*n-124)*x-(88*n+88)*x^2-(16*n+16)*x^3)*P(n-1,x) + (84*n^2+84*n-231+(80*n^2+80*n-556)*x+(16*n^2+16*n-448)*x^2-144*x^3-16*x^4)*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[((40*n^3 + 120*n^2 + 86*n + 6 + (16*n^3 + 48*n^2 - 92*n - 124)*x - (88*n + 88)*x^2 - (16*n + 16)*x^3)*P[n - 1, x] + (84*n^2 + 84*n - 231 + (80*n^2 + 80*n - 556)*x + (16*n^2 + 16*n - 448)*x^2 - 144*x^3 - 16*x^4)*D[P[n - 1, x], x])/(16*n^3 + 20*n - 32*n*x - 16*n*x^2)]];
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] = ((40*n^3 + 120*n^2 + 86*n + 6 + (16*n^3 + 48*n^2 - 92*n - 124)*x - (88*n + 88)*x^2 - (16*n + 16)*x^3)*v[i - 1] + (84*n^2 + 84*n - 231 + (80*n^2 + 80*n - 556)*x + (16*n^2 + 16*n - 448)*x^2 - 144*x^3 - 16*x^4)*deriv(v[i - 1], x))/(16*n^3 + 20*n - 32*n*x - 16*n*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

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  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.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023