A124848: Triangle read by rows: T(n,k) = (k+1)(k+2)(k+3)*binomial(n,k)/6 (0 <= k <= n).

A124848 ↗ nonnegative coefficients

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

Recurrence from A124848

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

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

\(P_{1}(x) = 1 + 4\,x\)

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	4														5
2	1	8	10													19
3	1	12	30	20												63
4	1	16	60	80	35											192
5	1	20	100	200	175	56										552
6	1	24	150	400	525	336	84									1520
7	1	28	210	700	1225	1176	588	120								4048
8	1	32	280	1120	2450	3136	2352	960	165							10496
9	1	36	360	1680	4410	7056	7056	4320	1485	220						26624
10	1	40	450	2400	7350	14112	17640	14400	7425	2200	286					66304
11	1	44	550	3300	11550	25872	38808	39600	27225	12100	3146	364				162560
12	1	48	660	4400	17325	44352	77616	95040	81675	48400	18876	4368	455			393216
13	1	52	780	5720	25025	72072	144144	205920	212355	157300	81796	28392	5915	560		940032
14	1	56	910	7280	35035	112112	252252	411840	495495	440440	286286	132496	41405	7840	680	2224128

b-files: A124848 ?

Technical details

Node id: n-7731fc957f5878fb · revision 0 · JSON
Definition source: oeis_catalog:A124848
Fingerprint: f6-f91847f627145760

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 ?

A124848 triangle 60 terms, 10 rows · tabl · nonn · catalog b-file?

Row sums ? A049612

Submission text ?

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

%e Example

Triangle begins:
  0: 1;
  1: 1,  4;
  2: 1,  8,  10;
  3: 1, 12,  30,   20;
  4: 1, 16,  60,   80,   35;
  5: 1, 20, 100,  200,  175,   56;
  6: 1, 24, 150,  400,  525,  336,   84;
  7: 1, 28, 210,  700, 1225, 1176,  588,  120;
  8: 1, 32, 280, 1120, 2450, 3136, 2352,  960,  165;
  9: 1, 36, 360, 1680, 4410, 7056, 7056, 4320, 1485, 220;

%F Formula

The row polynomials satisfy n*P(n,x) = (2*n-1+(n+3)*x)*P(n-1,x) - (n-1+(n-1)*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 1 + (n + 3)*x)*P[n - 1, x] - (n - 1 + (n - 1)*x)*P[n - 2, x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 1 + (n + 3)*x)*v[i - 1] - (n - 1 + (n - 1)*x)*v[i - 2])/(n)); 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

A124848 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0: 1;
  1: 1,  4;
  2: 1,  8,  10;
  3: 1, 12,  30,   20;
  4: 1, 16,  60,   80,   35;
  5: 1, 20, 100,  200,  175,   56;
  6: 1, 24, 150,  400,  525,  336,   84;
  7: 1, 28, 210,  700, 1225, 1176,  588,  120;
  8: 1, 32, 280, 1120, 2450, 3136, 2352,  960,  165;
  9: 1, 36, 360, 1680, 4410, 7056, 7056, 4320, 1485, 220;

%F Formula

The row polynomials satisfy n*P(n,x) = (2*n-1+(n+3)*x)*P(n-1,x) - (n-1+(n-1)*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 1 + (n + 3)*x)*P[n - 1, x] - (n - 1 + (n - 1)*x)*P[n - 2, x])/(n)]];
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; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 1 + (n + 3)*x)*v[i - 1] - (n - 1 + (n - 1)*x)*v[i - 2])/(n)); 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 Row-hypergeometric rows (multiplier sequences)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Row-hypergeometric rows (multiplier sequences)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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

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.