A269945: Triangle read by rows. Stirling set numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) = T(n-1, k-1) + k^2*T(n-1, k), for 0 <= k <= n.

A269945 ↗ nonnegative coefficients

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

Recurrence from A269945

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	0	1														1
3	0	1	1													2
4	0	1	5	1												7
5	0	1	21	14	1											37
6	0	1	85	147	30	1										264
7	0	1	341	1408	627	55	1									2433
8	0	1	1365	13013	11440	2002	91	1								27913
9	0	1	5461	118482	196053	61490	5278	140	1							386906
10	0	1	21845	1071799	3255330	1733303	251498	12138	204	1						6346119
11	0	1	87381	9668036	53157079	46587905	10787231	846260	25194	285	1					121159373
12	0	1	349525	87099705	860181300	1217854704	434928221	52253971	2458676	48279	385	1				2655174768
13	0	1	1398101	784246870	13850000505	31306548900	16875270660	2995372800	209609235	6369275	86779	506	1			66028903633
14	0	1	5592405	7059619931	222384254950	796513723005	638816292660	163648537860	16410363840	725520510	15047175	148005	650	1		1845579100993
15	0	1	22369621	63542171784	3565207699131	20135227330075	23793900258765	8657594647800	1213911823620	75177525150	2230238010	32955780	241605	819	1	57506847262162

b-files: A008957 (reversed triangle), A036969, A135920 (row sums), A269945 ?

Technical details

Node id: n-0164479e47952770 · revision 0 · JSON
Definition source: oeis_catalog:A269945
Fingerprint: f6-de3a1cd8e58b208d

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 ?

A036969 triangle 50 terms, 9 rows · tabl · nonn · catalog b-file?
A008957 reversed triangle 50 terms, 9 rows · tabl · nonn · catalog b-file?
A269945 triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?

Row sums ? A135920

Submission text ?

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

%e Example

Triangle begins:
   1: 1;
   2: 0, 1;
   3: 0, 1,     1;
   4: 0, 1,     5,       1;
   5: 0, 1,    21,      14,       1;
   6: 0, 1,    85,     147,      30,       1;
   7: 0, 1,   341,    1408,     627,      55,      1;
   8: 0, 1,  1365,   13013,   11440,    2002,     91,     1;
   9: 0, 1,  5461,  118482,  196053,   61490,   5278,   140,   1;
  10: 0, 1, 21845, 1071799, 3255330, 1733303, 251498, 12138, 204, 1;

%F Formula

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

A008957 reversed triangle, offset 1, from P₂, low zeros dropped

%e Example

Triangle begins:
   1: 1;
   2: 1,   1;
   3: 1,   5,     1;
   4: 1,  14,    21,      1;
   5: 1,  30,   147,     85,        1;
   6: 1,  55,   627,   1408,      341,        1;
   7: 1,  91,  2002,  11440,    13013,     1365,        1;
   8: 1, 140,  5278,  61490,   196053,   118482,     5461,       1;
   9: 1, 204, 12138, 251498,  1733303,  3255330,  1071799,   21845,     1;
  10: 1, 285, 25194, 846260, 10787231, 46587905, 53157079, 9668036, 87381, 1;

%F Formula

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

%t Mathematica

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

%o PARI/GP

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

A036969 triangle, offset 1, from P₂, low zeros dropped

%e Example

Triangle begins:
   1: 1;
   2: 1,     1;
   3: 1,     5,       1;
   4: 1,    21,      14,        1;
   5: 1,    85,     147,       30,        1;
   6: 1,   341,    1408,      627,       55,        1;
   7: 1,  1365,   13013,    11440,     2002,       91,      1;
   8: 1,  5461,  118482,   196053,    61490,     5278,    140,     1;
   9: 1, 21845, 1071799,  3255330,  1733303,   251498,  12138,   204,   1;
  10: 1, 87381, 9668036, 53157079, 46587905, 10787231, 846260, 25194, 285, 1;

%F Formula

Row n lists the coefficients of P(n,x) in ascending powers of x, starting at the lowest nonzero coefficient, where P(n,x) = x*P(n-1,x) + x*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = x.

%t Mathematica

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

%o PARI/GP

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

A269945 triangle, offset 0, from P₁

%e Example

Triangle begins:
  0: 1;
  1: 0, 1;
  2: 0, 1,     1;
  3: 0, 1,     5,       1;
  4: 0, 1,    21,      14,       1;
  5: 0, 1,    85,     147,      30,       1;
  6: 0, 1,   341,    1408,     627,      55,      1;
  7: 0, 1,  1365,   13013,   11440,    2002,     91,     1;
  8: 0, 1,  5461,  118482,  196053,   61490,   5278,   140,   1;
  9: 0, 1, 21845, 1071799, 3255330, 1733303, 251498, 12138, 204, 1;

%F Formula

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