A124848 → basis A007318

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one sign Hadamard kernel

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(4 + 4\,x + n\right)\,P_{n-1}(x) + \left(2 + 3\,x + x^{2}\right)\,P_{n-1}'(x)\)

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

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	4														9
2	19	28	10													57
3	63	132	90	20												305
4	192	516	510	220	35											1473
5	552	1800	2310	1460	455	56										6633
6	1520	5808	9120	7540	3465	840	84									28377
7	4048	17696	32760	33320	20125	7224	1428	120								116721
8	10496	51584	109760	132160	98560	46648	13692	2280	165							465345
9	26624	145152	348480	483840	428400	250992	97356	24120	3465	220						1808649
10	66304	396800	1059840	1664640	1703520	1187424	571200	187320	40095	5060	286					6882489
11	162560	1058816	3111680	5448960	6320160	5100480	2923536	1190640	337755	63580	7150	364				25725681
12	393216	2767872	8870400	17121280	22176000	20313216	13497792	6557760	2312640	577500	96954	9828	455			94694913
13	940032	7108608	24660480	51983360	74314240	76108032	57465408	32398080	13642200	4238520	944658	143052	13195	560		343960425
14	2224128	17977344	67092480	153287680	239599360	271118848	229092864	146889600	71831760	26666640	7399392	1488396	205205	17360	680	1234891737

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(\left(n\right)\,P_{n}(x) = \left(2 + x - 2\,n - n x\right)\,P_{n-2}(x) + \left(2 + 3\,x + 3\,n + n x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 5 + 4\,x\)

Technical details

Node id: n-60da2299a7c8ee9b · revision 0 · JSON
Definition source: derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-17a210f1346c4343)
Fingerprint: f6-b7ef7f83c60f9896

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:     5,      4;
  2:    19,     28,     10;
  3:    63,    132,     90,     20;
  4:   192,    516,    510,    220,     35;
  5:   552,   1800,   2310,   1460,    455,     56;
  6:  1520,   5808,   9120,   7540,   3465,    840,    84;
  7:  4048,  17696,  32760,  33320,  20125,   7224,  1428,   120;
  8: 10496,  51584, 109760, 132160,  98560,  46648, 13692,  2280,  165;
  9: 26624, 145152, 348480, 483840, 428400, 250992, 97356, 24120, 3465, 220;

%F Formula

The row polynomials satisfy n*P(n,x) = (n+4+4*x)*P(n-1,x) + (2+3*x+x^2)*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[((n + 4 + 4*x)*P[n - 1, x] + (2 + 3*x + x^2)*D[P[n - 1, x], x])/(n)]];
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] = ((n + 4 + 4*x)*v[i - 1] + (2 + 3*x + x^2)*deriv(v[i - 1], x))/(n)); 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 (−∞, -1]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle read by rows: T(n,k) = (k+1)*(k+2)*(k+3)*binomial(n,k)/6 (0 <= k <= n).?
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
Ore-algebra identity of two definitions polytool ? step › direct check
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle read by rows: T(n,k) = (k+1)*(k+2)*(k+3)*binomial(n,k)/6 (0 <= k <= n).?
  - ↑ Row-hypergeometric rows (multiplier sequences) (above)

References

[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.