A071951: Triangle of Legendre-Stirling numbers of the second kind T(n,j), n >= 1, 1 <= j <= n, read by rows.

A071951 ↗ nonnegative coefficients

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

Recurrence from A071951

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	2	1														3
3	4	8	1													13
4	8	52	20	1												81
5	16	320	292	40	1											669
6	32	1936	3824	1092	70	1										6955
7	64	11648	47824	25664	3192	112	1									88505
8	128	69952	585536	561104	121424	7896	168	1								1346209
9	256	419840	7096384	11807616	4203824	453056	17304	240	1							23998521
10	512	2519296	85576448	243248704	137922336	23232176	1422080	34584	330	1						493956467
11	1024	15116288	1029436672	4950550528	4380918784	1113673728	102868656	3912128	64284	440	1					11596542533
12	2048	90698752	12368356352	100040447232	136378114048	51155215360	6874318464	384541872	9697688	112684	572	1				307301505073
13	4096	544194560	148510974976	2013177300992	4191383868672	2284897159168	436117049344	34561333248	1257333792	22092928	188188	728	1			9110471500693
14	8192	3265171456	1782675894272	40412056994816	127754693361152	100157064553728	26707451922432	2924533043200	147721374528	3687555872	46933744	301756	910	1		299893197116059
15	16384	19591036928	21395375902720	810023815790592	3873052857829376	4334351404617728	1595774372209920	237273831032832	16219456750720	553352520448	9882810080	94007680	467376	1120	1	10888674034993905

b-files: A071951, A191935 (reversed triangle) ?

Technical details

Node id: n-9401e1a9db1eb1a5 · revision 0 · JSON
Definition source: oeis_catalog:A071951
Fingerprint: f6-57cb71c89ecc0cb7

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 ?

A071951 triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?
A191935 reversed triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?

Row sums ? A135921

Submission text ?

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

%e Example

Triangle begins:
   1:   1;
   2:   2,       1;
   3:   4,       8,        1;
   4:   8,      52,       20,         1;
   5:  16,     320,      292,        40,         1;
   6:  32,    1936,     3824,      1092,        70,        1;
   7:  64,   11648,    47824,     25664,      3192,      112,       1;
   8: 128,   69952,   585536,    561104,    121424,     7896,     168,     1;
   9: 256,  419840,  7096384,  11807616,   4203824,   453056,   17304,   240,   1;
  10: 512, 2519296, 85576448, 243248704, 137922336, 23232176, 1422080, 34584, 330, 1;

%F Formula

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

A071951 triangle, offset 1, from P₁

%e Example

Triangle begins:
   1:   1;
   2:   2,       1;
   3:   4,       8,        1;
   4:   8,      52,       20,         1;
   5:  16,     320,      292,        40,         1;
   6:  32,    1936,     3824,      1092,        70,        1;
   7:  64,   11648,    47824,     25664,      3192,      112,       1;
   8: 128,   69952,   585536,    561104,    121424,     7896,     168,     1;
   9: 256,  419840,  7096384,  11807616,   4203824,   453056,   17304,   240,   1;
  10: 512, 2519296, 85576448, 243248704, 137922336, 23232176, 1422080, 34584, 330, 1;

%F Formula

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

A191935 reversed triangle, offset 1, from P₁

%e Example

Triangle begins:
   1: 1;
   2: 1,   2;
   3: 1,   8,     4;
   4: 1,  20,    52,       8;
   5: 1,  40,   292,     320,       16;
   6: 1,  70,  1092,    3824,     1936,        32;
   7: 1, 112,  3192,   25664,    47824,     11648,        64;
   8: 1, 168,  7896,  121424,   561104,    585536,     69952,      128;
   9: 1, 240, 17304,  453056,  4203824,  11807616,   7096384,   419840,     256;
  10: 1, 330, 34584, 1422080, 23232176, 137922336, 243248704, 85576448, 2519296, 512;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (2+x)*P(n-1,x) + 4*x*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[(2 + x)*P[n - 1, x] + 4*x*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];
Table[Reverse[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] = (2 + x)*v[i - 1] + 4*x*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 = Vecrev(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

Used in 7

Related ?

Same fingerprint ?

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.