A357367: Triangle read by rows. T(n, k) = binomial(n - 1, k - 1)*(n + k)! / k!.

A357367 ↗ nonnegative coefficients

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

Recurrence from A357367

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	0	2							2
2	0	6	12						18
3	0	24	120	120					264
4	0	120	1080	2520	1680				5400
5	0	720	10080	40320	60480	30240			141840
6	0	5040	100800	604800	1512000	1663200	665280		4551120
7	0	40320	1088640	9072000	33264000	59875200	51891840	17297280	172529280

All 15 rows b-files: A032037 (row sums), A357367 ?

Technical details

Node id: n-4cba07695b3e3f90 · revision 1 · JSON
Definition source: oeis_catalog:A357367
Fingerprint: f6-1e5fbd8fe1b88630

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 ?

A357367 triangle 36 terms, 8 rows · tabl · nonn · catalog b-file?

Row sums ? A032037

Submission text ?

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

%e Example

Triangle begins:
  0: 1;
  1: 0,       2;
  2: 0,       6,        12;
  3: 0,      24,       120,        120;
  4: 0,     120,      1080,       2520,        1680;
  5: 0,     720,     10080,      40320,       60480,       30240;
  6: 0,    5040,    100800,     604800,     1512000,     1663200,       665280;
  7: 0,   40320,   1088640,    9072000,    33264000,    59875200,     51891840,     17297280;
  8: 0,  362880,  12700800,  139708800,   698544000,  1816214400,   2542700160,   1816214400,   518918400;
  9: 0, 3628800, 159667200, 2235340800, 14529715200, 50854003200, 101708006400, 116237721600, 70572902400, 17643225600;

%F Formula

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

%t Mathematica

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

A357367 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0: 1;
  1: 0,       2;
  2: 0,       6,        12;
  3: 0,      24,       120,        120;
  4: 0,     120,      1080,       2520,        1680;
  5: 0,     720,     10080,      40320,       60480,       30240;
  6: 0,    5040,    100800,     604800,     1512000,     1663200,       665280;
  7: 0,   40320,   1088640,    9072000,    33264000,    59875200,     51891840,     17297280;
  8: 0,  362880,  12700800,  139708800,   698544000,  1816214400,   2542700160,   1816214400,   518918400;
  9: 0, 3628800, 159667200, 2235340800, 14529715200, 50854003200, 101708006400, 116237721600, 70572902400, 17643225600;

%F Formula

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

%t Mathematica

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