A105278: Triangle read by rows: T(n,k) = binomial(n,k)*(n-1)!/(k-1)!.

A105278 ↗ nonnegative coefficients

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

Recurrence from A105278

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

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

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	6	6	1													13
4	24	36	12	1												73
5	120	240	120	20	1											501
6	720	1800	1200	300	30	1										4051
7	5040	15120	12600	4200	630	42	1									37633
8	40320	141120	141120	58800	11760	1176	56	1								394353
9	362880	1451520	1693440	846720	211680	28224	2016	72	1							4596553
10	3628800	16329600	21772800	12700800	3810240	635040	60480	3240	90	1						58941091
11	39916800	199584000	299376000	199584000	69854400	13970880	1663200	118800	4950	110	1					824073141
12	479001600	2634508800	4390848000	3293136000	1317254400	307359360	43908480	3920400	217800	7260	132	1				12470162233
13	6227020800	37362124800	68497228800	57081024000	25686460800	6849722880	1141620480	122316480	8494200	377520	10296	156	1			202976401213
14	87178291200	566658892800	1133317785600	1038874636800	519437318400	155831195520	29682132480	3710266560	309188880	17177160	624624	14196	182	1		3535017524403
15	1307674368000	9153720576000	19833061248000	19833061248000	10908183686400	3636061228800	779155977600	111307996800	10821610800	721440720	32792760	993720	19110	210	1	65573803186921

b-files: A089231 (reversed triangle), A105278 ?

Technical details

Node id: n-4a0fd05ab0621175 · revision 1 · JSON
Definition source: oeis_catalog:A105278
Fingerprint: f6-8fea465cea2922a0

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 ?

A105278 triangle 47 terms, 9 rows · tabl · nonn · catalog b-file?
A089231 reversed triangle 45 terms, 9 rows · tabl · nonn · catalog b-file?

Row sums ? A000262

Submission text ?

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

%e Example

Triangle begins:
   1:       1;
   2:       2,        1;
   3:       6,        6,        1;
   4:      24,       36,       12,        1;
   5:     120,      240,      120,       20,       1;
   6:     720,     1800,     1200,      300,      30,      1;
   7:    5040,    15120,    12600,     4200,     630,     42,     1;
   8:   40320,   141120,   141120,    58800,   11760,   1176,    56,    1;
   9:  362880,  1451520,  1693440,   846720,  211680,  28224,  2016,   72,  1;
  10: 3628800, 16329600, 21772800, 12700800, 3810240, 635040, 60480, 3240, 90, 1;

%F Formula

The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) + (2+2*x)*d/dx P(n-1,x) + x*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] + (2 + 2*x)*D[P[n - 1, x], x] + x*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] + (2 + 2*x)*deriv(v[i - 1], x) + 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, v[i])))};
T(10)

polytool c48629374597

A089231 reversed triangle, offset 1, from P₁

%e Example

Triangle begins:
   1: 1;
   2: 1,  2;
   3: 1,  6,    6;
   4: 1, 12,   36,    24;
   5: 1, 20,  120,   240,    120;
   6: 1, 30,  300,  1200,   1800,     720;
   7: 1, 42,  630,  4200,  12600,   15120,     5040;
   8: 1, 56, 1176, 11760,  58800,  141120,   141120,    40320;
   9: 1, 72, 2016, 28224, 211680,  846720,  1693440,  1451520,   362880;
  10: 1, 90, 3240, 60480, 635040, 3810240, 12700800, 21772800, 16329600, 3628800;

%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) + (2+2*x)*d/dx P(n-1,x) + x*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] + (2 + 2*x)*D[P[n - 1, x], x] + x*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] + (2 + 2*x)*deriv(v[i - 1], x) + x*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

A105278 triangle, offset 1, from P₁

%e Example

Triangle begins:
   1:       1;
   2:       2,        1;
   3:       6,        6,        1;
   4:      24,       36,       12,        1;
   5:     120,      240,      120,       20,       1;
   6:     720,     1800,     1200,      300,      30,      1;
   7:    5040,    15120,    12600,     4200,     630,     42,     1;
   8:   40320,   141120,   141120,    58800,   11760,   1176,    56,    1;
   9:  362880,  1451520,  1693440,   846720,  211680,  28224,  2016,   72,  1;
  10: 3628800, 16329600, 21772800, 12700800, 3810240, 635040, 60480, 3240, 90, 1;

%F Formula

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

Proof chains ? JSON

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

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.