A151775: Triangle read by rows: T(n,k) = value of (d^2n/dx^2n) (tan^(2k)(x)/cos(x)) at the point x = 0.

A151775 ↗ nonnegative coefficients

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

Recurrence from A151775

\(P_{n}(x) = \left(1 + 2\,x\right)\,P_{n-1}(x) + \left(2 + 12\,x + 10\,x^{2}\right)\,P_{n-1}'(x) + \left(4\,x + 8\,x^{2} + 4\,x^{3}\right)\,P_{n-1}''(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	2														3
2	5	28	24													57
3	61	662	1320	720												2763
4	1385	24568	83664	100800	40320											250737
5	50521	1326122	6749040	13335840	11491200	3628800										36581523
6	2702765	98329108	692699304	1979524800	2739623040	1836172800	479001600									7828053417
7	199360981	9596075582	88966364760	342459477360	673354281600	715588473600	392302310400	87178291200								2309644635483
8	19391512145	1192744081648	14036558182944	69485886288000	181217064188160	270530842982400	233114750668800	108101081088000	20922789888000							898621108880097
9	2404879675441	183983154281042	2674388403120480	16475552674505280	54515459807827200	106867999164326400	128349384991027200	92958909332889600	37347179950080000	6402373705728000						445777636063460643
10	370371188237525	34489251602450188	606124302614470584	4533519378664886400	18445492546696615680	45466338016836172800	71198454890468505600	71458612085532672000	44623477666639872000	15813863053148160000	2432902008176640000					274613643571568682777
11	69348874393137901	7722592644581974502	161270548786369288200	1436480458879959939600	7020443925991498464000	21106351076465332339200	41292140398505963366400	53751829118068092211200	46308904323109908480000	25411783120505413632000	8055338549072855040000	1124000727777607680000				205676334188681975553003
12	15514534163557086905	2035778987564783402728	49799177378374922698224	520083815323911587496000	2997978871847924944425600	10746906073840860473088000	25493311186314183184896000	41258826648781130188800000	45914797479305898823680000	34638624809065525248000000	16950305641795583016960000	4860179146910375608320000	620448401733239439360000			184053312545818735778213457
13	4087072509293123892361	624086284447168370787962	17668710080081496856339920	213587916447367529505940320	1431053176721765608419408000	6010001417378615589957024000	16907104453141498613681664000	33029300050079452866819072000	45525102213503236830105600000	44218502166759811381370880000	29663287238067578442547200000	13096890200086518015590400000	3427977419576147902464000000	403291461126605635584000000		193944394596325636374396208563
14	1252259641403629865468285	220145816803809737344107268	7139543642117733276737601864	98824012096847558985181972800	760363761509783997969278977920	3688742886960310767144228672000	12096576490041726745717759104000	27908452862795979243846236160000	46269074381406835043143618560000	55492404412028141074419302400000	47800618663023533118305894400000	28863832012280897632410009600000	11608623721091005238594764800000	2794809825607377054597120000000	304888344611713860501504000000	237694572315172714282278845824137

b-files: A000281 (row sums), A151775 ?

Technical details

Node id: n-3338feb205e3b77f · revision 0 · JSON
Definition source: oeis_catalog:A151775
Fingerprint: f6-a766906061ecf2ab

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 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 ?

A151775 triangle 30 terms, 7 rows · tabl · nonn · catalog b-file?

Row sums ? A000281

Submission text ?

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

%e Example

Triangle begins:
  0:             1;
  1:             1,               2;
  2:             5,              28,               24;
  3:            61,             662,             1320,               720;
  4:          1385,           24568,            83664,            100800,             40320;
  5:         50521,         1326122,          6749040,          13335840,          11491200,            3628800;
  6:       2702765,        98329108,        692699304,        1979524800,        2739623040,         1836172800,          479001600;
  7:     199360981,      9596075582,      88966364760,      342459477360,      673354281600,       715588473600,       392302310400,       87178291200;
  8:   19391512145,   1192744081648,   14036558182944,    69485886288000,   181217064188160,    270530842982400,    233114750668800,   108101081088000,    20922789888000;
  9: 2404879675441, 183983154281042, 2674388403120480, 16475552674505280, 54515459807827200, 106867999164326400, 128349384991027200, 92958909332889600, 37347179950080000, 6402373705728000;

%F Formula

The row polynomials satisfy P(n,x) = (1+2*x)*P(n-1,x) + (2+12*x+10*x^2)*d/dx P(n-1,x) + (4*x+8*x^2+4*x^3)*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[(1 + 2*x)*P[n - 1, x] + (2 + 12*x + 10*x^2)*D[P[n - 1, x], x] + (4*x + 8*x^2 + 4*x^3)*D[P[n - 1, x], {x, 2}]];
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)*v[i - 1] + (2 + 12*x + 10*x^2)*deriv(v[i - 1], x) + (4*x + 8*x^2 + 4*x^3)*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 - 1, v[i])))};
T(9)

polytool c48629374597

A151775 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0:             1;
  1:             1,               2;
  2:             5,              28,               24;
  3:            61,             662,             1320,               720;
  4:          1385,           24568,            83664,            100800,             40320;
  5:         50521,         1326122,          6749040,          13335840,          11491200,            3628800;
  6:       2702765,        98329108,        692699304,        1979524800,        2739623040,         1836172800,          479001600;
  7:     199360981,      9596075582,      88966364760,      342459477360,      673354281600,       715588473600,       392302310400,       87178291200;
  8:   19391512145,   1192744081648,   14036558182944,    69485886288000,   181217064188160,    270530842982400,    233114750668800,   108101081088000,    20922789888000;
  9: 2404879675441, 183983154281042, 2674388403120480, 16475552674505280, 54515459807827200, 106867999164326400, 128349384991027200, 92958909332889600, 37347179950080000, 6402373705728000;

%F Formula

The row polynomials satisfy P(n,x) = (1+2*x)*P(n-1,x) + (2+12*x+10*x^2)*d/dx P(n-1,x) + (4*x+8*x^2+4*x^3)*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[(1 + 2*x)*P[n - 1, x] + (2 + 12*x + 10*x^2)*D[P[n - 1, x], x] + (4*x + 8*x^2 + 4*x^3)*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] = (1 + 2*x)*v[i - 1] + (2 + 12*x + 10*x^2)*deriv(v[i - 1], x) + (4*x + 8*x^2 + 4*x^3)*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 ≥ 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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 1
by 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.