A127952: Triangle read by rows, T(n,k) = (n+1)*binomial(n-1,k-1).

A127952 ↗ palindromicγ-positivenonnegative coefficientsonly rational roots

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

Recurrence from A127952

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

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

\(P_{1}(x) = 2\,x\)

\(P_{2}(x) = 3\,x + 3\,x^{2}\)

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	2														2
2	0	3	3													6
3	0	4	8	4												16
4	0	5	15	15	5											40
5	0	6	24	36	24	6										96
6	0	7	35	70	70	35	7									224
7	0	8	48	120	160	120	48	8								512
8	0	9	63	189	315	315	189	63	9							1152
9	0	10	80	280	560	700	560	280	80	10						2560
10	0	11	99	396	924	1386	1386	924	396	99	11					5632
11	0	12	120	540	1440	2520	3024	2520	1440	540	120	12				12288
12	0	13	143	715	2145	4290	6006	6006	4290	2145	715	143	13			26624
13	0	14	168	924	3080	6930	11088	12936	11088	6930	3080	924	168	14		57344
14	0	15	195	1170	4290	10725	19305	25740	25740	19305	10725	4290	1170	195	15	122880

b-files: A127952 ?

Technical details

Node id: n-a88a7983f557eac4 · revision 1 · JSON
Definition source: oeis_catalog:A127952
Fingerprint: f6-77a7ed3b1d5f257e

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: yes
γ-positive (gamma_positive): yes
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

Matches ?

A127952 triangle 66 terms, 11 rows · tabl · nonn · catalog b-file?

Row sums ? A057711, A129952

Submission text ?

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

%e Example

Triangle begins:
  0: 1;
  1: 0,  2;
  2: 0,  3,  3;
  3: 0,  4,  8,   4;
  4: 0,  5, 15,  15,   5;
  5: 0,  6, 24,  36,  24,   6;
  6: 0,  7, 35,  70,  70,  35,   7;
  7: 0,  8, 48, 120, 160, 120,  48,   8;
  8: 0,  9, 63, 189, 315, 315, 189,  63,  9;
  9: 0, 10, 80, 280, 560, 700, 560, 280, 80, 10;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (2*n-4+(2*n-4)*x)*P(n-1,x) - (n-2+(2*n-4)*x+(n-2)*x^2)*P(n-2,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2*x, P(2,x) = 3*x + 3*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2*x;
P[2, x_] := 3*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 4 + (2*n - 4)*x)*P[n - 1, x] - (n - 2 + (2*n - 4)*x + (n - 2)*x^2)*P[n - 2, x])/(n - 2)]];
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] = 2*x; v[3] = 3*x + 3*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((2*n - 4 + (2*n - 4)*x)*v[i - 1] - (n - 2 + (2*n - 4)*x + (n - 2)*x^2)*v[i - 2])/(n - 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

A127952 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0: 1;
  1: 0,  2;
  2: 0,  3,  3;
  3: 0,  4,  8,   4;
  4: 0,  5, 15,  15,   5;
  5: 0,  6, 24,  36,  24,   6;
  6: 0,  7, 35,  70,  70,  35,   7;
  7: 0,  8, 48, 120, 160, 120,  48,   8;
  8: 0,  9, 63, 189, 315, 315, 189,  63,  9;
  9: 0, 10, 80, 280, 560, 700, 560, 280, 80, 10;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (2*n-4+(2*n-4)*x)*P(n-1,x) - (n-2+(2*n-4)*x+(n-2)*x^2)*P(n-2,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2*x, P(2,x) = 3*x + 3*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2*x;
P[2, x_] := 3*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 4 + (2*n - 4)*x)*P[n - 1, x] - (n - 2 + (2*n - 4)*x + (n - 2)*x^2)*P[n - 2, x])/(n - 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; v[2] = 2*x; v[3] = 3*x + 3*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((2*n - 4 + (2*n - 4)*x)*v[i - 1] - (n - 2 + (2*n - 4)*x + (n - 2)*x^2)*v[i - 2])/(n - 2)); 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 Row-hypergeometric rows (multiplier sequences), Liu–Wang theorem for three-term recurrences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Row-hypergeometric rows (multiplier sequences), Liu–Wang theorem for three-term recurrences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Row-hypergeometric rows (multiplier sequences), Liu–Wang theorem for three-term recurrences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 2
by Liu–Wang theorem for three-term recurrences

Proof chains ? JSON

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
Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

Related ?

Same fingerprint ?

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.