A304334: T(n, k) = Sum_{j=0..k} (-1)^jbinomial(2k, j)(k - j)^(2n)/k!, triangle read by rows, n >= 0 and 0 <= k <= n.

A304334 ↗ nonnegative coefficients

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

Recurrence from A304334

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	0	1														1
2	0	1	6													7
3	0	1	30	60												91
4	0	1	126	840	840											1807
5	0	1	510	8820	25200	15120										49651
6	0	1	2046	84480	526680	831600	332640									1777447
7	0	1	8190	780780	9609600	30270240	30270240	8648640								79587691
8	0	1	32766	7108920	164684520	929728800	1755673920	1210809600	259459200							4327497727
9	0	1	131070	64307940	2734477200	26207541360	83658294720	104977192320	52929676800	8821612800						279393234211
10	0	1	524286	580082160	44651946360	704409123600	3588264519840	7318998086400	6536815084800	2514159648000	335221286400					21043100301847
11	0	1	2097150	5225982300	722552292000	18413963124480	144674523433440	451925783749440	637926108019200	425898644371200	129060195264000	14079294028800				1822706292362011
12	0	1	8388606	47054812200	11634000424200	473355019368000	5613390032342400	25905901012992000	54385044425712000	56187277866720000	29090168012505600	7124122778572800	647647525324800			179438587737162607
13	0	1	33554430	423577195860	186802774158000	12043287491835600	212495851590422400	1415337290477510400	4257819873635328000	6400261017678528000	5044133360185920000	2083805912732544000	420970891461120000	32382376266240000		19879437087904356691
14	0	1	134217726	3812530307040	2994774467270040	304444637230734000	7914802982075589600	74876419374748992000	314960590626986304000	663187018135561920000	747623254690376064000	463994116568446464000	156474880356098304000	26521166162050560000	1748648318376960000	2457608340439083686407

b-files: A304334 ?

Technical details

Node id: n-01512c1ccffbc04e · revision 0 · JSON
Definition source: oeis_catalog:A304334
Fingerprint: f6-56207361c0b00b39

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 ?

A304334 triangle 45 terms, 9 rows · tabl · nonn · catalog b-file?

Submission text ?

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

%e Example

Triangle begins:
  0: 1;
  1: 0, 1;
  2: 0, 1,      6;
  3: 0, 1,     30,       60;
  4: 0, 1,    126,      840,        840;
  5: 0, 1,    510,     8820,      25200,       15120;
  6: 0, 1,   2046,    84480,     526680,      831600,      332640;
  7: 0, 1,   8190,   780780,    9609600,    30270240,    30270240,      8648640;
  8: 0, 1,  32766,  7108920,  164684520,   929728800,  1755673920,   1210809600,   259459200;
  9: 0, 1, 131070, 64307940, 2734477200, 26207541360, 83658294720, 104977192320, 52929676800, 8821612800;

%F Formula

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

%t Mathematica

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

%o PARI/GP

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

A304334 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0: 1;
  1: 0, 1;
  2: 0, 1,      6;
  3: 0, 1,     30,       60;
  4: 0, 1,    126,      840,        840;
  5: 0, 1,    510,     8820,      25200,       15120;
  6: 0, 1,   2046,    84480,     526680,      831600,      332640;
  7: 0, 1,   8190,   780780,    9609600,    30270240,    30270240,      8648640;
  8: 0, 1,  32766,  7108920,  164684520,   929728800,  1755673920,   1210809600,   259459200;
  9: 0, 1, 131070, 64307940, 2734477200, 26207541360, 83658294720, 104977192320, 52929676800, 8821612800;

%F Formula

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

%t Mathematica

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

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 2)*x*v[i - 1] + ((n - 1)*x + (4*n - 4)*x^2)*deriv(v[i - 1], x) + (n - 1)*x^2*deriv(deriv(v[i - 1], x), x))/(n - 1)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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 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
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Iterated real-rootedness preservers

Proof chains ? JSON

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

Used in 1

derivative of T(n, k) = Sum_{j=0..k} (-1)^j*binomial(2*k, j)*(k - j)^(2*n)/k!, … Image under a real-rootedness preserver (derivative polynomial, shift 0)

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.