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	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	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
8	725760	5806080	13547520	13547520	6773760	1806336	258048	18432	512							42483968
9	7257600	65318400	174182400	203212800	121927680	40642560	7741440	829440	46080	1024						621159424
10	79833600	798336000	2395008000	3193344000	2235340800	894136320	212889600	30412800	2534400	112640	2048					9841950208
11	958003200	10538035200	35126784000	52690176000	42152140800	19670999040	5620285440	1003622400	111513600	7434240	270336	4096				167879268352
12	12454041600	149448499200	547977830400	913296384000	821966745600	438382264320	146127421440	31313018880	4349030400	386580480	21086208	638976	8192			3065723549696
13	174356582400	2266635571200	9066542284800	16621994188800	16621994188800	9973196513280	3799312957440	949828239360	158304706560	17589411840	1279229952	58146816	1490944	16384		59651093528576
14	2615348736000	36614882304000	158664489984000	317328979968000	349061877964800	232707918643200	99731965132800	28494847180800	5540664729600	738755297280	67159572480	4070277120	156549120	3440640	32768	1231571119812608

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.