A397048: Triangle T(n, k) read by rows: T(n, k) = (Sum_{i=0..k} (-1)^i * binomial(k, i) * (n+i)^n) * (-1)^k / k! for 0 <= k <= n.

A397048 ↗ nonnegative coefficients

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

Recurrence from A397048

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	1							2
2	4	5	1						10
3	27	37	12	1					77
4	256	369	151	22	1				799
5	3125	4651	2190	425	35	1			10427
6	46656	70993	36751	8550	965	51	1		163967
7	823543	1273609	706104	186501	25830	1904	70	1	3017562

All 15 rows b-files: A134980 (row sums), A397048 ?

Technical details

Node id: n-735e311cbe9a718e · revision 0 · JSON
Definition source: oeis_catalog:A397048
Fingerprint: f6-f13747014c7999ad

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 ?

A397048 triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?

Row sums ? A134980

Submission text ?

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

%e Example

Triangle begins:
  0:         1;
  1:         1,         1;
  2:         4,         5,         1;
  3:        27,        37,        12,         1;
  4:       256,       369,       151,        22,        1;
  5:      3125,      4651,      2190,       425,       35,       1;
  6:     46656,     70993,     36751,      8550,      965,      51,      1;
  7:    823543,   1273609,    706104,    186501,    25830,    1904,     70,    1;
  8:  16777216,  26269505,  15341887,   4453638,   714021,   65562,   3402,   92,   1;
  9: 387420489, 612579511, 372684090, 116419465, 20853525, 2245467, 146664, 5646, 117, 1;

%F Formula

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

A397048 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0:         1;
  1:         1,         1;
  2:         4,         5,         1;
  3:        27,        37,        12,         1;
  4:       256,       369,       151,        22,        1;
  5:      3125,      4651,      2190,       425,       35,       1;
  6:     46656,     70993,     36751,      8550,      965,      51,      1;
  7:    823543,   1273609,    706104,    186501,    25830,    1904,     70,    1;
  8:  16777216,  26269505,  15341887,   4453638,   714021,   65562,   3402,   92,   1;
  9: 387420489, 612579511, 372684090, 116419465, 20853525, 2245467, 146664, 5646, 117, 1;

%F Formula

The row polynomials satisfy P(n,x) = (n+x)*P(n-1,x) + (n+2*x)*d/dx P(n-1,x) + x*d^2/dx^2 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[(n + x)*P[n - 1, x] + (n + 2*x)*D[P[n - 1, x], x] + x*D[P[n - 1, x], {x, 2}]];
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; for(n = 1, N, my(i = n + 1); v[i] = (n + x)*v[i - 1] + (n + 2*x)*deriv(v[i - 1], x) + x*deriv(deriv(v[i - 1], x), x)); 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

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.