A176230: Exponential Riordan array [1/sqrt(1-2x), x/(1-2x)].

A176230 ↗ nonnegative coefficients

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

Recurrence from A176230

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	1							2
2	3	6	1						10
3	15	45	15	1					76
4	105	420	210	28	1				764
5	945	4725	3150	630	45	1			9496
6	10395	62370	51975	13860	1485	66	1		140152
7	135135	945945	945945	315315	45045	3003	91	1	2390480

All 15 rows b-files: A066223 (row sums), A119743 (reversed triangle), A176230 ?

Technical details

Node id: n-7d781fe2b71cdf22 · revision 1 · JSON
Definition source: oeis_catalog:A176230
Fingerprint: f6-86230d7253d1861d

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 ?

A176230 triangle 46 terms, 9 rows · tabl · nonn · catalog b-file?
A119743 reversed triangle 47 terms, 8 rows · tabl · nonn b-file?

Row sums ? A066223

Submission text ?

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

%e Example

Triangle begins:
  0:        1;
  1:        1,         1;
  2:        3,         6,         1;
  3:       15,        45,        15,         1;
  4:      105,       420,       210,        28,        1;
  5:      945,      4725,      3150,       630,       45,       1;
  6:    10395,     62370,     51975,     13860,     1485,      66,      1;
  7:   135135,    945945,    945945,    315315,    45045,    3003,     91,    1;
  8:  2027025,  16216200,  18918900,   7567560,  1351350,  120120,   5460,  120,   1;
  9: 34459425, 310134825, 413513100, 192972780, 41351310, 4594590, 278460, 9180, 153, 1;

%F Formula

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

A119743 reversed triangle, offset 1, from P₁

%e Example

Triangle begins:
   1: 1,   1;
   2: 1,   6,     3;
   3: 1,  15,    45,     15;
   4: 1,  28,   210,    420,      105;
   5: 1,  45,   630,   3150,     4725,       945;
   6: 1,  66,  1485,  13860,    51975,     62370,      10395;
   7: 1,  91,  3003,  45045,   315315,    945945,     945945,     135135;
   8: 1, 120,  5460, 120120,  1351350,   7567560,   18918900,   16216200,    2027025;
   9: 1, 153,  9180, 278460,  4594590,  41351310,  192972780,  413513100,  310134825,   34459425;
  10: 1, 190, 14535, 581400, 13226850, 174594420, 1309458150, 5237832600, 9820936125, 6547290750, 654729075;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+x)*P(n-1,x) + (2+4*x)*d/dx P(n-1,x) + 4*x*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1 + x.

%t Mathematica

P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x] + (2 + 4*x)*D[P[n - 1, x], x] + 4*x*D[P[n - 1, x], {x, 2}]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1 + x; for(n = 2, N, my(i = n); v[i] = (1 + x)*v[i - 1] + (2 + 4*x)*deriv(v[i - 1], x) + 4*x*deriv(deriv(v[i - 1], x), x)); 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, v[i])))};
T(10)

polytool c48629374597

A176230 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0:        1;
  1:        1,         1;
  2:        3,         6,         1;
  3:       15,        45,        15,         1;
  4:      105,       420,       210,        28,        1;
  5:      945,      4725,      3150,       630,       45,       1;
  6:    10395,     62370,     51975,     13860,     1485,      66,      1;
  7:   135135,    945945,    945945,    315315,    45045,    3003,     91,    1;
  8:  2027025,  16216200,  18918900,   7567560,  1351350,  120120,   5460,  120,   1;
  9: 34459425, 310134825, 413513100, 192972780, 41351310, 4594590, 278460, 9180, 153, 1;

%F Formula

The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + (2+4*x)*d/dx P(n-1,x) + 4*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[(1 + x)*P[n - 1, x] + (2 + 4*x)*D[P[n - 1, x], x] + 4*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] = (1 + x)*v[i - 1] + (2 + 4*x)*deriv(v[i - 1], x) + 4*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, Row-hypergeometric rows (multiplier sequences)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Iterated real-rootedness preservers, Row-hypergeometric rows (multiplier sequences)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 1
by Iterated real-rootedness preservers
Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0)
by Row-hypergeometric rows (multiplier sequences)

Proof chains ? JSON

Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
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

Used in 6

Related ?

Same fingerprint ?

A176230 → reverse

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.