A089503: Triangle of numbers used for basis change between certain falling factorials.

A089503 ↗ nonnegative coefficients

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

Recurrence from A089503

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	1	4														5
3	1	12	30													43
4	1	24	168	336												529
5	1	40	540	2880	5040											8501
6	1	60	1320	13200	59400	95040										169021
7	1	84	2730	43680	360360	1441440	2162160									4010455
8	1	112	5040	117600	1528800	11007360	40360320	57657600								110676833
9	1	144	8568	274176	5140800	57576960	374250240	1283143680	1764322560							3484717129
10	1	180	13680	574560	14651280	234420480	2344204800	14065228800	45711993600	60949324800						123320412181
11	1	220	20790	1108800	36867600	796340160	11281485600	103145011200	580190688000	1805037696000	2346549004800					4847038223171
12	1	264	30360	2003760	84157920	2356421760	44772013440	575640172800	4892941468800	26095687833600	78287063500800	99638080819200				209536628422705
13	1	312	42900	3432000	177606000	6251731200	153167414400	2625727104000	31180509360000	249444074880000	1272164781888000	3700843001856000	4626053752320000			9882471447634813
14	1	364	58968	5621616	351351000	15178363200	465469804800	10240335705600	161285287363200	1792058748480000	13619646488448000	66860082761472000	189436901157504000	233153109116928000		505033804901100749
15	1	420	79170	8867040	658377720	34235641440	1283836554000	35213802624000	708677777808000	10393940741184000	109136377782432000	793719111144960000	3770165777938560000	10440459077368320000	12677700308232960000	27802319803528367791

b-files: A089503 ?

Technical details

Node id: n-ee693a689121a12c · revision 0 · JSON
Definition source: oeis_catalog:A089503
Fingerprint: f6-fe953b9def4ff3da

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 ?

A089503 triangle 42 terms, 8 rows · tabl · nonn · catalog b-file?

Row sums ? A251568

Submission text ?

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

%e Example

Triangle begins:
   1: 1;
   2: 1,   4;
   3: 1,  12,    30;
   4: 1,  24,   168,    336;
   5: 1,  40,   540,   2880,     5040;
   6: 1,  60,  1320,  13200,    59400,     95040;
   7: 1,  84,  2730,  43680,   360360,   1441440,    2162160;
   8: 1, 112,  5040, 117600,  1528800,  11007360,   40360320,    57657600;
   9: 1, 144,  8568, 274176,  5140800,  57576960,  374250240,  1283143680,  1764322560;
  10: 1, 180, 13680, 574560, 14651280, 234420480, 2344204800, 14065228800, 45711993600, 60949324800;

%F Formula

The row polynomials satisfy P(n,x) = (1+(4*n-4)*x+(n^2-3*n+2)*x^2)*P(n-1,x) - (x^2+(n-1)*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 + (4*n - 4)*x + (n^2 - 3*n + 2)*x^2)*P[n - 1, x] - (x^2 + (n - 1)*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 + (4*n - 4)*x + (n^2 - 3*n + 2)*x^2)*v[i - 1] - (x^2 + (n - 1)*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

A089503 triangle, offset 1, from P₁

%e Example

Triangle begins:
   1: 1;
   2: 1,   4;
   3: 1,  12,    30;
   4: 1,  24,   168,    336;
   5: 1,  40,   540,   2880,     5040;
   6: 1,  60,  1320,  13200,    59400,     95040;
   7: 1,  84,  2730,  43680,   360360,   1441440,    2162160;
   8: 1, 112,  5040, 117600,  1528800,  11007360,   40360320,    57657600;
   9: 1, 144,  8568, 274176,  5140800,  57576960,  374250240,  1283143680,  1764322560;
  10: 1, 180, 13680, 574560, 14651280, 234420480, 2344204800, 14065228800, 45711993600, 60949324800;

%F Formula

The row polynomials satisfy P(n,x) = (1+(4*n-4)*x+(n^2-3*n+2)*x^2)*P(n-1,x) - (x^2+(n-1)*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 + (4*n - 4)*x + (n^2 - 3*n + 2)*x^2)*P[n - 1, x] - (x^2 + (n - 1)*x^3)*D[P[n - 1, x], x]];
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] = (1 + (4*n - 4)*x + (n^2 - 3*n + 2)*x^2)*v[i - 1] - (x^2 + (n - 1)*x^3)*deriv(v[i - 1], 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 Row-hypergeometric rows (multiplier sequences)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Row-hypergeometric rows (multiplier sequences)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from 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

Used in 7

Related ?

Same fingerprint ?

A089503 → reverse

References

[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.