A272866 → basis A013611 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(5 - 16\,x^{2} + 20\,n x + 32\,n x^{2}\right)\,P_{n}(x) = \left(25 + 200\,x + 560\,x^{2} + 640\,x^{3} + 256\,x^{4} + 100\,n x + 560\,n x^{2} + 960\,n x^{3} + 512\,n x^{4}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	5	40	48													93
2	25	400	1680	2560	1280											5945
3	125	3000	21600	70400	115200	92160	28672									331157
4	625	20000	204000	1024000	2912000	4915200	4874240	2621440	589824							17161329
5	3125	125000	1650000	11200000	45600000	119040000	204288000	229376000	162201600	65536000	11534336					850554061
6	15625	750000	12150000	104000000	552000000	1958400000	4820480000	8355840000	10174464000	8519680000	4671406080	1509949440	218103808			40897238953
7	78125	4375000	84000000	868000000	5684000000	25536000000	82163200000	193740800000	338042880000	435814400000	409757286400	273049190400	122138132480	32883343360	4026531840	1923792217605

All 15 rows

Technical details

Node id: n-b552d7a4ad6358e9 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-6e3a14f88852ba15

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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:       5,        40,         48;
  2:      25,       400,       1680,        2560,         1280;
  3:     125,      3000,      21600,       70400,       115200,         92160,          28672;
  4:     625,     20000,     204000,     1024000,      2912000,       4915200,        4874240,        2621440,          589824;
  5:    3125,    125000,    1650000,    11200000,     45600000,     119040000,      204288000,      229376000,       162201600,        65536000,        11534336;
  6:   15625,    750000,   12150000,   104000000,    552000000,    1958400000,     4820480000,     8355840000,     10174464000,      8519680000,      4671406080,      1509949440,       218103808;
  7:   78125,   4375000,   84000000,   868000000,   5684000000,   25536000000,    82163200000,   193740800000,    338042880000,    435814400000,    409757286400,    273049190400,    122138132480,     32883343360,      4026531840;
  8:  390625,  25000000,  555000000,  6720000000,  52360000000,  284928000000,  1133977600000,  3391488000000,   7739596800000,  13565952000000,  18247319552000,  18673041408000,  14274894233600,   7892002406400,   2979633561600,    687194767360,    73014444032;
  9: 1953125, 140625000, 3543750000, 49200000000, 444600000000, 2842560000000, 13491072000000, 48955392000000, 138334003200000, 307650560000000, 541039656960000, 751954821120000, 820997259264000, 695482712064000, 447549014016000, 211312390963200, 68998649610240, 13915694039040, 1305670057984;

%F Formula

The row polynomials satisfy (5+20*n*x+(32*n-16)*x^2)*P(n,x) = (25+(100*n+200)*x+(560*n+560)*x^2+(960*n+640)*x^3+(512*n+256)*x^4)*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[((25 + (100*n + 200)*x + (560*n + 560)*x^2 + (960*n + 640)*x^3 + (512*n + 256)*x^4)*P[n - 1, x])/(5 + 20*n*x + (32*n - 16)*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] = ((25 + (100*n + 200)*x + (560*n + 560)*x^2 + (960*n + 640)*x^3 + (512*n + 256)*x^4)*v[i - 1])/(5 + 20*n*x + (32*n - 16)*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

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from basis_map_A013611 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Parent-row provers of polytool (above)

References

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