A304330: T(n, k) = Sum_{j=0..k} (-1)^jbinomial(2k, j)(k - j)^(2n), triangle read by rows, n >= 0 and 0 <= k <= n.

A304330 ↗ nonnegative coefficients

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

Recurrence from A304330

\(\left(-1 + n\right)\,P_{n}(x) = \left(-2\,x + 2\,n x\right)\,P_{n-1}(x) + \left(-x - 10\,x^{2} + n x + 10\,n x^{2}\right)\,P_{n-1}'(x) + \left(-x^{2} - 4\,x^{3} + n x^{2} + 4\,n x^{3}\right)\,P_{n-1}''(x)\)

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

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

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	12													13
3	0	1	60	360												421
4	0	1	252	5040	20160											25453
5	0	1	1020	52920	604800	1814400										2473141
6	0	1	4092	506880	12640320	99792000	239500800									352444093
7	0	1	16380	4684680	230630400	3632428800	21794572800	43589145600								69251478661
8	0	1	65532	42653520	3952428480	111567456000	1264085222400	6102480384000	10461394944000							17943523153933
9	0	1	262140	385847640	65627452800	3144904963200	60233972198400	529085049292800	2134124568576000	3201186852864000						5927841361456981
10	0	1	1048572	3480492960	1071646712640	84529094832000	2583550454284800	36887750355456000	263564384219136000	912338253066240000	1216451004088320000					2431910546406522973
11	0	1	4194300	31355893800	17341255008000	2209675574937600	104165656872076800	2277705950097177600	25721180675334144000	154550100069421056000	468333636574003200000	562000363888803840000				1212989379862721528101
12	0	1	16777212	282328873200	279216010180800	56802602324160000	4041640823286528000	130565741105479680000	2192804991244707840000	20389239392275353600000	105562401683780321280000	284372184127734743040000	310224200866619719680000			722875495525684291639213
13	0	1	67108860	2541463175160	4483266579792000	1445194499020272000	152997013145104128000	7133299944006652416000	171675297304976424960000	2322526718095184240640000	18304151137442666496000000	83178863857362412339200000	201645730563302817792000000	201645730563302817792000000		507275965883448333971692021
14	0	1	268435452	22875181842240	71874587214480960	36533356467688080000	5698658147094424512000	377377153648734919680000	12699211014080087777280000	240657305141032709529600000	2712975266620436661043200000	18521160352239363814195200000	74951718050379657373286400000	165147853331345007771648000000	152444172305856930250752000000	414031618935013558427928710653

b-files: A100872 (row sums), A304330 ?

Technical details

Node id: n-177268549ede7415 · revision 0 · JSON
Definition source: oeis_catalog:A304330
Fingerprint: f6-4ffec972130b4198

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 ?

A304330 triangle 45 terms, 9 rows · tabl · nonn · catalog b-file?

Row sums ? A100872

Submission text ?

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

%e Example

Triangle begins:
  0: 1;
  1: 0, 1;
  2: 0, 1,     12;
  3: 0, 1,     60,       360;
  4: 0, 1,    252,      5040,       20160;
  5: 0, 1,   1020,     52920,      604800,       1814400;
  6: 0, 1,   4092,    506880,    12640320,      99792000,      239500800;
  7: 0, 1,  16380,   4684680,   230630400,    3632428800,    21794572800,     43589145600;
  8: 0, 1,  65532,  42653520,  3952428480,  111567456000,  1264085222400,   6102480384000,   10461394944000;
  9: 0, 1, 262140, 385847640, 65627452800, 3144904963200, 60233972198400, 529085049292800, 2134124568576000, 3201186852864000;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (2*n-2)*x*P(n-1,x) + ((n-1)*x+(10*n-10)*x^2)*d/dx P(n-1,x) + ((n-1)*x^2+(4*n-4)*x^3)*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 + (10*n - 10)*x^2)*D[P[n - 1, x], x] + ((n - 1)*x^2 + (4*n - 4)*x^3)*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 + (10*n - 10)*x^2)*deriv(v[i - 1], x) + ((n - 1)*x^2 + (4*n - 4)*x^3)*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

A304330 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0: 1;
  1: 0, 1;
  2: 0, 1,     12;
  3: 0, 1,     60,       360;
  4: 0, 1,    252,      5040,       20160;
  5: 0, 1,   1020,     52920,      604800,       1814400;
  6: 0, 1,   4092,    506880,    12640320,      99792000,      239500800;
  7: 0, 1,  16380,   4684680,   230630400,    3632428800,    21794572800,     43589145600;
  8: 0, 1,  65532,  42653520,  3952428480,  111567456000,  1264085222400,   6102480384000,   10461394944000;
  9: 0, 1, 262140, 385847640, 65627452800, 3144904963200, 60233972198400, 529085049292800, 2134124568576000, 3201186852864000;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (2*n-2)*x*P(n-1,x) + ((n-1)*x+(10*n-10)*x^2)*d/dx P(n-1,x) + ((n-1)*x^2+(4*n-4)*x^3)*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 + (10*n - 10)*x^2)*D[P[n - 1, x], x] + ((n - 1)*x^2 + (4*n - 4)*x^3)*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 + (10*n - 10)*x^2)*deriv(v[i - 1], x) + ((n - 1)*x^2 + (4*n - 4)*x^3)*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), tri… 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.