A376827: T(n, k) = binomial(n, k)*hypergeom([(1 - n)/2, -n/2], [1], 4).

A376827 ↗ palindromicγ-positivenonnegative coefficientsonly rational roots

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

Recurrence from A376827

\(\left(n\right)\,P_{n}(x) = \left(-1 - x + 2\,n + 2\,n x\right)\,P_{n-1}(x) + \left(-3 - 6\,x - 3\,x^{2} + 3\,n + 6\,n x + 3\,n x^{2}\right)\,P_{n-2}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	1							2
2	3	6	3						12
3	7	21	21	7					56
4	19	76	114	76	19				304
5	51	255	510	510	255	51			1632
6	141	846	2115	2820	2115	846	141		9024
7	393	2751	8253	13755	13755	8253	2751	393	50304

All 15 rows b-files: A098453 (row sums), A376827, A376827 (reversed triangle) ?

Technical details

Node id: n-63b9a6ee72661305 · revision 0 · JSON
Definition source: oeis_catalog:A376827
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 ?

A376827 triangle 45 terms, 9 rows · tabl · nonn · catalog b-file?
A376827 reversed triangle 45 terms, 9 rows · tabl · nonn · catalog b-file?

Row sums ? A098453

Submission text ?

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

%e Example

Triangle begins:
  0:    1;
  1:    1,     1;
  2:    3,     6,      3;
  3:    7,    21,     21,      7;
  4:   19,    76,    114,     76,     19;
  5:   51,   255,    510,    510,    255,     51;
  6:  141,   846,   2115,   2820,   2115,    846,    141;
  7:  393,  2751,   8253,  13755,  13755,   8253,   2751,    393;
  8: 1107,  8856,  30996,  61992,  77490,  61992,  30996,   8856,  1107;
  9: 3139, 28251, 113004, 263676, 395514, 395514, 263676, 113004, 28251, 3139;

%F Formula

The row polynomials satisfy n*P(n,x) = (2*n-1+(2*n-1)*x)*P(n-1,x) + (3*n-3+(6*n-6)*x+(3*n-3)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 1 + (2*n - 1)*x)*P[n - 1, x] + (3*n - 3 + (6*n - 6)*x + (3*n - 3)*x^2)*P[n - 2, x])/(n)]];
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] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 1 + (2*n - 1)*x)*v[i - 1] + (3*n - 3 + (6*n - 6)*x + (3*n - 3)*x^2)*v[i - 2])/(n)); 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

A376827 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0:    1;
  1:    1,     1;
  2:    3,     6,      3;
  3:    7,    21,     21,      7;
  4:   19,    76,    114,     76,     19;
  5:   51,   255,    510,    510,    255,     51;
  6:  141,   846,   2115,   2820,   2115,    846,    141;
  7:  393,  2751,   8253,  13755,  13755,   8253,   2751,    393;
  8: 1107,  8856,  30996,  61992,  77490,  61992,  30996,   8856,  1107;
  9: 3139, 28251, 113004, 263676, 395514, 395514, 263676, 113004, 28251, 3139;

%F Formula

The row polynomials satisfy n*P(n,x) = (2*n-1+(2*n-1)*x)*P(n-1,x) + (3*n-3+(6*n-6)*x+(3*n-3)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 1 + (2*n - 1)*x)*P[n - 1, x] + (3*n - 3 + (6*n - 6)*x + (3*n - 3)*x^2)*P[n - 2, x])/(n)]];
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] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 1 + (2*n - 1)*x)*v[i - 1] + (3*n - 3 + (6*n - 6)*x + (3*n - 3)*x^2)*v[i - 2])/(n)); 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

A376827 reversed triangle, offset 0, from P₀

%e Example

Triangle begins:
  0:    1;
  1:    1,     1;
  2:    3,     6,      3;
  3:    7,    21,     21,      7;
  4:   19,    76,    114,     76,     19;
  5:   51,   255,    510,    510,    255,     51;
  6:  141,   846,   2115,   2820,   2115,    846,    141;
  7:  393,  2751,   8253,  13755,  13755,   8253,   2751,    393;
  8: 1107,  8856,  30996,  61992,  77490,  61992,  30996,   8856,  1107;
  9: 3139, 28251, 113004, 263676, 395514, 395514, 263676, 113004, 28251, 3139;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where n*P(n,x) = (2*n-1+(2*n-1)*x)*P(n-1,x) + (3*n-3+(6*n-6)*x+(3*n-3)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 1 + (2*n - 1)*x)*P[n - 1, x] + (3*n - 3 + (6*n - 6)*x + (3*n - 3)*x^2)*P[n - 2, x])/(n)]];
Table[Reverse[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] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 1 + (2*n - 1)*x)*v[i - 1] + (3*n - 3 + (6*n - 6)*x + (3*n - 3)*x^2)*v[i - 2])/(n)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(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)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Row-hypergeometric rows (multiplier sequences)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Row-hypergeometric rows (multiplier sequences)

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

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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.