A094262: Triangle read by rows: T(n,k) is the number of rooted trees with k nodes which are disjoint sets of labels with union {1..n}. If a node has an empty set of labels then it must have at least two children.

A094262 ↗ nonnegative coefficients

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

Recurrence from A094262

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	x²⁴	x²⁵	x²⁶	x²⁷	x²⁸	P_n(1)
0	1																													1
1	1	2	1																											4
2	1	6	12	10	3																									32
3	1	14	61	124	131	70	15																							416
4	1	30	240	890	1830	2226	1600	630	105																					7552
5	1	62	841	5060	16990	35216	47062	40796	22225	6930	945																			176128
6	1	126	2772	25410	127953	401436	836976	1196532	1182195	795718	349020	90090	10395																	5018624
7	1	254	8821	118524	851361	3789786	11315829	23698536	35643099	38808946	30393335	16716700	6136515	1351350	135135															168968192
8	1	510	27480	527530	5231460	31726422	128945040	370748610	779175270	1219253530	1431293496	1257343230	815302180	379068690	119639520	22972950	2027025													6563282944
9	1	1022	84481	2276020	30459980	244783252	1309489644	4961577972	13847170470	29183679040	47141876486	58769110972	56519394316	41592937100	23007444460	9263714460	2564186625	436486050	34459425											288909131776
10	1	2046	257532	9613010	170761503	1782403392	12256126432	59494178304	213176209290	580778482636	1226858756376	2034187404756	2663771430782	2757034703280	2244466031808	1420743884560	685141702245	243258525510	59959399500	9166207050	654729075									14212910809088
11	1	4094	780781	40001324	931484191	12440487422	108035533243	658233616832	2956285621498	10120910020012	27029601856466	57208926323704	96944972634446	132266791903324	145492990769126	128664368953120	90778203721205	50405373488470	21543944447025	6840434781180	1519626183075	210822762150	13749310575							772776684683264
12	1	8190	2358720	164698170	4979773890	84117771738	910192002720	6853009324590	37894557195495	159688110321740	526350391414848	1382367456926820	2930624496437820	5058811491026580	7146149679077760	8275154712970236	7842884131297215	6052385552563350	3767157853338880	1862775307403010	714864485585250	205271096080050	41504585522400	5270569053750	316234143225					46017323176296448
13	1	16382	7108921	672961380	26223530970	555163162008	7405656149346	68071469265228	457070507971935	2333807207333030	9324667526757583	29764999197465896	77064791725498668	163608007264877280	286937085388602300	417610326654228696	505326226233266607	508003475817769386	422790909337771975	289350444521077940	161099165706849370	71795778656401800	24998687891827650	6551668705081500	1215498635175825	142305364451250	7905853580625			2978458881388183552
14	1	32766	21392292	2734531810	136522672653	3595906127268	58632668644368	651581919184716	5254617673036485	32109912970946954	153361121512741428	585660748031587878	1818773615964997601	4651376676940838040	9886922776677081024	17580389889834010056	26258132579729885091	33006919483118572722	34911698252822897740	30997126009976742798	22990127538104621895	14132680980550071940	7117066562237551920	2886686822386867500	919837385235343575	221714920156479750	38000802877537500	4126855569086250	213458046676875	208198894960190160896

b-files: A005172 (row sums), A094262 ?

Technical details

Node id: n-f712c326ca8e5415 · revision 0 · JSON
Definition source: oeis_catalog:A094262
Fingerprint: f6-5870953131fc996b

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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 ?

A094262 triangle 49 terms, 7 rows · tabf · nonn · catalog b-file?

Row sums ? A005172, A225170

Submission text ?

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

%e Example

Triangle begins:
  0: 1;
  1: 1,    2,     1;
  2: 1,    6,    12,      10,        3;
  3: 1,   14,    61,     124,      131,        70,         15;
  4: 1,   30,   240,     890,     1830,      2226,       1600,        630,         105;
  5: 1,   62,   841,    5060,    16990,     35216,      47062,      40796,       22225,        6930,         945;
  6: 1,  126,  2772,   25410,   127953,    401436,     836976,    1196532,     1182195,      795718,      349020,       90090,       10395;
  7: 1,  254,  8821,  118524,   851361,   3789786,   11315829,   23698536,    35643099,    38808946,    30393335,    16716700,     6136515,     1351350,      135135;
  8: 1,  510, 27480,  527530,  5231460,  31726422,  128945040,  370748610,   779175270,  1219253530,  1431293496,  1257343230,   815302180,   379068690,   119639520,   22972950,    2027025;
  9: 1, 1022, 84481, 2276020, 30459980, 244783252, 1309489644, 4961577972, 13847170470, 29183679040, 47141876486, 58769110972, 56519394316, 41592937100, 23007444460, 9263714460, 2564186625, 436486050, 34459425;

%F Formula

The row polynomials satisfy P(n,x) = (1+2*x+x^2)*P(n-1,x) + (x+2*x^2+x^3)*d/dx 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 + 2*x + x^2)*P[n - 1, x] + (x + 2*x^2 + x^3)*D[P[n - 1, x], x]];
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 + 2*x + x^2)*v[i - 1] + (x + 2*x^2 + x^3)*deriv(v[i - 1], 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

A094262 triangle, offset 1, from P₀

%e Example

Triangle begins:
   1: 1;
   2: 1,    2,     1;
   3: 1,    6,    12,      10,        3;
   4: 1,   14,    61,     124,      131,        70,         15;
   5: 1,   30,   240,     890,     1830,      2226,       1600,        630,         105;
   6: 1,   62,   841,    5060,    16990,     35216,      47062,      40796,       22225,        6930,         945;
   7: 1,  126,  2772,   25410,   127953,    401436,     836976,    1196532,     1182195,      795718,      349020,       90090,       10395;
   8: 1,  254,  8821,  118524,   851361,   3789786,   11315829,   23698536,    35643099,    38808946,    30393335,    16716700,     6136515,     1351350,      135135;
   9: 1,  510, 27480,  527530,  5231460,  31726422,  128945040,  370748610,   779175270,  1219253530,  1431293496,  1257343230,   815302180,   379068690,   119639520,   22972950,    2027025;
  10: 1, 1022, 84481, 2276020, 30459980, 244783252, 1309489644, 4961577972, 13847170470, 29183679040, 47141876486, 58769110972, 56519394316, 41592937100, 23007444460, 9263714460, 2564186625, 436486050, 34459425;

%F Formula

The row polynomials satisfy P(n,x) = (1+2*x+x^2)*P(n-1,x) + (x+2*x^2+x^3)*d/dx 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 + 2*x + x^2)*P[n - 1, x] + (x + 2*x^2 + x^3)*D[P[n - 1, x], x]];
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 + 2*x + x^2)*v[i - 1] + (x + 2*x^2 + x^3)*deriv(v[i - 1], 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

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
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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

Used in 7

Related ?

Same fingerprint ?

A094262 → 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.