A086915: Triangle read by rows: T(n,k) = 2^k * (n!/k!)*binomial(n-1,k-1).

A086915 ↗ nonnegative coefficients

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

Recurrence from A086915

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

\(P_{0}(x) = 2\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	2								2
1	4	4							8
2	12	24	8						44
3	48	144	96	16					304
4	240	960	960	320	32				2512
5	1440	7200	9600	4800	960	64			24064
6	10080	60480	100800	67200	20160	2688	128		261536
7	80640	564480	1128960	940800	376320	75264	7168	256	3173888

All 15 rows b-files: A086915 ?

Technical details

Node id: n-cbd7c2e66114e926 · revision 1 · JSON
Definition source: oeis_catalog:A086915
Fingerprint: f6-269631c8dfab89fe

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 ?

A086915 triangle 42 terms, 8 rows · tabl · nonn · catalog b-file?

Row sums ? A052897

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:       2;
  1:       4,        4;
  2:      12,       24,         8;
  3:      48,      144,        96,        16;
  4:     240,      960,       960,       320,        32;
  5:    1440,     7200,      9600,      4800,       960,       64;
  6:   10080,    60480,    100800,     67200,     20160,     2688,     128;
  7:   80640,   564480,   1128960,    940800,    376320,    75264,    7168,    256;
  8:  725760,  5806080,  13547520,  13547520,   6773760,  1806336,  258048,  18432,   512;
  9: 7257600, 65318400, 174182400, 203212800, 121927680, 40642560, 7741440, 829440, 46080, 1024;

%F Formula

The row polynomials satisfy 2*P(n,x) = (4+4*x)*P(n-1,x) + (2+4*x)*d/dx P(n-1,x) + x*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 2.

%t Mathematica

P[0, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((4 + 4*x)*P[n - 1, x] + (2 + 4*x)*D[P[n - 1, x], x] + x*D[P[n - 1, x], {x, 2}])/(2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 2; for(n = 1, N, my(i = n + 1); v[i] = ((4 + 4*x)*v[i - 1] + (2 + 4*x)*deriv(v[i - 1], x) + x*deriv(deriv(v[i - 1], x), 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

A086915 triangle, offset 1, from P₀

%e Example

Triangle begins:
   1:       2;
   2:       4,        4;
   3:      12,       24,         8;
   4:      48,      144,        96,        16;
   5:     240,      960,       960,       320,        32;
   6:    1440,     7200,      9600,      4800,       960,       64;
   7:   10080,    60480,    100800,     67200,     20160,     2688,     128;
   8:   80640,   564480,   1128960,    940800,    376320,    75264,    7168,    256;
   9:  725760,  5806080,  13547520,  13547520,   6773760,  1806336,  258048,  18432,   512;
  10: 7257600, 65318400, 174182400, 203212800, 121927680, 40642560, 7741440, 829440, 46080, 1024;

%F Formula

The row polynomials satisfy 2*P(n,x) = (4+4*x)*P(n-1,x) + (2+4*x)*d/dx P(n-1,x) + x*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 2.

%t Mathematica

P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((4 + 4*x)*P[n - 1, x] + (2 + 4*x)*D[P[n - 1, x], x] + x*D[P[n - 1, x], {x, 2}])/(2)]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 2; for(n = 2, N, my(i = n); v[i] = ((4 + 4*x)*v[i - 1] + (2 + 4*x)*deriv(v[i - 1], x) + x*deriv(deriv(v[i - 1], x), x))/(2)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Iterated real-rootedness preservers
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Iterated real-rootedness preservers, Row-hypergeometric rows (multiplier sequences)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Iterated real-rootedness preservers, Row-hypergeometric rows (multiplier sequences)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 1
by Iterated real-rootedness preservers
Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0)
by Row-hypergeometric rows (multiplier sequences)

Proof chains ? JSON

Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
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

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.