Real-rooted atlas

A013612 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(9 + 150\,x + 400\,x^{2} + 90\,n x + 400\,n x^{2} + 100\,n^{2} x^{2}\right)\,P_{n}(x) = \left(54 + 1620\,x + 10200\,x^{2} + 18000\,x^{3} + 540\,n x + 5400\,n x^{2} + 12000\,n x^{3} + 600\,n^{2} x^{2} + 2000\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 36 + 960\,x + 3600\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
03696036004596
1216864064800128000201656
2129669120777600307200040000007920016
37776518400777600046080000120000000115200000289582176
44665637324806998400055296000021600000004147200000313600000010069923136
5279936261273605878656005806080000302400000008709120000013171200000081920000000337383552896
61679616179159040470292480055738368000362880000000139345920000031610880000003932160000000207360000000010983809331456
7100776961209323520362797056005016453120003919104000000188116992000005689958400000010616832000000011197440000000051200000000000349512251618816

All 15 rows

Technical details
Node id
n-a6b0708a7afa5887 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-40433d4e786667ae

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:        36,         960,          3600;
  1:       216,        8640,         64800,         128000;
  2:      1296,       69120,        777600,        3072000,         4000000;
  3:      7776,      518400,       7776000,       46080000,       120000000,        115200000;
  4:     46656,     3732480,      69984000,      552960000,      2160000000,       4147200000,        3136000000;
  5:    279936,    26127360,     587865600,     5806080000,     30240000000,      87091200000,      131712000000,       81920000000;
  6:   1679616,   179159040,    4702924800,    55738368000,    362880000000,    1393459200000,     3161088000000,     3932160000000,     2073600000000;
  7:  10077696,  1209323520,   36279705600,   501645312000,   3919104000000,   18811699200000,    56899584000000,   106168320000000,   111974400000000,     51200000000000;
  8:  60466176,  8062156800,  272097792000,  4299816960000,  39191040000000,  225740390400000,   853493760000000,  2123366400000000,  3359232000000000,   3072000000000000,  1239040000000000;
  9: 362797056, 53210234880, 1995383808000, 35473489920000, 369515520000000, 2483144294400000, 11266117632000000, 35035545600000000, 73903104000000000, 101376000000000000, 81776640000000000, 29491200000000000;
%F Formula
The row polynomials satisfy (9+(90*n+150)*x+(100*n^2+400*n+400)*x^2)*P(n,x) = (54+(540*n+1620)*x+(600*n^2+5400*n+10200)*x^2+(2000*n^2+12000*n+18000)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 36 + 960*x + 3600*x^2.
%t Mathematica
P[0, x_] := 36 + 960*x + 3600*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((54 + (540*n + 1620)*x + (600*n^2 + 5400*n + 10200)*x^2 + (2000*n^2 + 12000*n + 18000)*x^3)*P[n - 1, x])/(9 + (90*n + 150)*x + (100*n^2 + 400*n + 400)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 36 + 960*x + 3600*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((54 + (540*n + 1620)*x + (600*n^2 + 5400*n + 10200)*x^2 + (2000*n^2 + 12000*n + 18000)*x^3)*v[i - 1])/(9 + (90*n + 150)*x + (100*n^2 + 400*n + 400)*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. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.