A272866 → basis A013611 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-15 - 12\,x + 16\,x^{2} + 20\,n + 32\,n x\right)\,P_{n}(x) = \left(25 + 200\,x + 560\,x^{2} + 640\,x^{3} + 256\,x^{4} + 100\,n + 560\,n x + 960\,n x^{2} + 512\,n x^{3}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	25	52	16													93
2	225	1320	2480	1664	256											5945
3	1625	15900	60000	109760	99840	39936	4096									331157
4	10625	146000	836000	2585600	4678400	4997120	2990080	851968	65536							17161329
5	65625	1162500	8950000	39280000	108320000	194944000	229888000	172851200	77004800	17039360	1048576					850554061
6	390625	8475000	82050000	467600000	1742400000	4458240000	8000000000	10088448000	8798208000	5098700800	1808793600	327155712	16777216			40897238953
7	2265625	58187500	679000000	4764200000	22416800000	74681600000	181260800000	324700160000	429793280000	416088064000	287545753600	135496990720	39929774080	6106906624	268435456	1923792217605

All 15 rows

Technical details

Node id: n-9cbd345833a8bbca · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-72d6e0ea460d9703

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:       25,         52,          16;
  2:      225,       1320,        2480,         1664,           256;
  3:     1625,      15900,       60000,       109760,         99840,          39936,           4096;
  4:    10625,     146000,      836000,      2585600,       4678400,        4997120,        2990080,          851968,           65536;
  5:    65625,    1162500,     8950000,     39280000,     108320000,      194944000,      229888000,       172851200,        77004800,        17039360,         1048576;
  6:   390625,    8475000,    82050000,    467600000,    1742400000,     4458240000,     8000000000,     10088448000,      8798208000,      5098700800,      1808793600,       327155712,        16777216;
  7:  2265625,   58187500,   679000000,   4764200000,   22416800000,    74681600000,   181260800000,    324700160000,    429793280000,    416088064000,    287545753600,    135496990720,     39929774080,      6106906624,       268435456;
  8: 12890625,  382500000,  5225000000,  43568000000,  247912000000,  1019468800000,  3129548800000,   7303577600000,  13069312000000,  17945067520000,  18775801856000,  14732912230400,   8426356736000,   3344705781760,    842887331840,   111669149696,     4294967296;
  9: 72265625, 2432812500, 38081250000, 367980000000, 2457720000000, 12037536000000, 44763264000000, 129082982400000, 292255948800000, 522619289600000, 738477342720000, 820506525696000, 708959010816000, 467389749657600, 227941967462400, 78146929950720, 17201344020480, 2010044694528, 68719476736;

%F Formula

The row polynomials satisfy (20*n-15+(32*n-12)*x+16*x^2)*P(n,x) = (100*n+25+(560*n+200)*x+(960*n+560)*x^2+(512*n+640)*x^3+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[((100*n + 25 + (560*n + 200)*x + (960*n + 560)*x^2 + (512*n + 640)*x^3 + 256*x^4)*P[n - 1, x])/(20*n - 15 + (32*n - 12)*x + 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] = ((100*n + 25 + (560*n + 200)*x + (960*n + 560)*x^2 + (512*n + 640)*x^3 + 256*x^4)*v[i - 1])/(20*n - 15 + (32*n - 12)*x + 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 differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] 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.