A156289: Triangle read by rows: T(n,k) is the number of end rhyme patterns of a poem of an even number of lines (2n) with 1<=k<=n evenly rhymed sounds.

A156289 ↗ nonnegative coefficients

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

Recurrence from A156289

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	1	3							4
3	1	15	15						31
4	1	63	210	105					379
5	1	255	2205	3150	945				6556
6	1	1023	21120	65835	51975	10395			150349
7	1	4095	195195	1201200	1891890	945945	135135		4373461
8	1	16383	1777230	20585565	58108050	54864810	18918900	2027025	156297964

All 15 rows b-files: A156289 ?

Technical details

Node id: n-67b93749ab954f15 · revision 0 · JSON
Definition source: oeis_catalog:A156289
Fingerprint: f6-7e5db8abc6982e99

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 -1)
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 ?

A156289 triangle 39 terms, 8 rows · tabl · nonn · catalog b-file?

Row sums ? A005046

Submission text ?

Atlas rows n = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 1;
   2: 1,      3;
   3: 1,     15,        15;
   4: 1,     63,       210,        105;
   5: 1,    255,      2205,       3150,         945;
   6: 1,   1023,     21120,      65835,       51975,        10395;
   7: 1,   4095,    195195,    1201200,     1891890,       945945,       135135;
   8: 1,  16383,   1777230,   20585565,    58108050,     54864810,     18918900,     2027025;
   9: 1,  65535,  16076985,  341809650,  1637971335,   2614321710,   1640268630,   413513100,   34459425;
  10: 1, 262143, 145020540, 5581493295, 44025570225, 112133266245, 114359345100, 51068867850, 9820936125, 654729075;

%F Formula

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

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + 3*x)*P[n - 1, x] + (3*x + 2*x^2)*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + 3*x)*v[i - 1] + (3*x + 2*x^2)*deriv(v[i - 1], x) + x^2*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, v[i])))};
T(10)

polytool c48629374597

A156289 triangle, offset 1, from P₁

%e Example

Triangle begins:
   1: 1;
   2: 1,      3;
   3: 1,     15,        15;
   4: 1,     63,       210,        105;
   5: 1,    255,      2205,       3150,         945;
   6: 1,   1023,     21120,      65835,       51975,        10395;
   7: 1,   4095,    195195,    1201200,     1891890,       945945,       135135;
   8: 1,  16383,   1777230,   20585565,    58108050,     54864810,     18918900,     2027025;
   9: 1,  65535,  16076985,  341809650,  1637971335,   2614321710,   1640268630,   413513100,   34459425;
  10: 1, 262143, 145020540, 5581493295, 44025570225, 112133266245, 114359345100, 51068867850, 9820936125, 654729075;

%F Formula

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

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + 3*x)*P[n - 1, x] + (3*x + 2*x^2)*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + 3*x)*v[i - 1] + (3*x + 2*x^2)*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
by Iterated real-rootedness preservers
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Iterated real-rootedness preservers
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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.