A153649: Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (j+1)prime(j)T(n-2, k-1) with j=4, read by rows.

A153649 ↗ palindromicγ-positivenonnegative coefficients

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

Recurrence from A153649

\(\left(-2 + n\right)\,P_{n}(x) = \left(-2 - 2\,x + n + n x\right)\,P_{n-1}(x) + \left(-70\,x + 35\,n x\right)\,P_{n-2}(x)\)

\(P_{0}(x) = 2\)

\(P_{1}(x) = 7 + 7\,x\)

\(P_{2}(x) = 2 + 94\,x + 2\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	2															2
1	7	7														14
2	2	94	2													98
3	2	341	341	2												686
4	2	413	3972	413	2											4802
5	2	485	16320	16320	485	2										33614
6	2	557	31260	171660	31260	557	2									235298
7	2	629	48792	774120	774120	48792	629	2								1647086
8	2	701	68916	1917012	7556340	1917012	68916	701	2							11529602
9	2	773	91632	3693648	36567552	36567552	3693648	91632	773	2						80707214
10	2	845	116940	6197340	107356620	337607004	107356620	6197340	116940	845	2					564950498
11	2	917	144840	9521400	242831640	1724827944	1724827944	242831640	9521400	144840	917	2				3954653486
12	2	989	175332	13759140	469259940	5725141284	15265901028	5725141284	469259940	13759140	175332	989	2			27682574402
13	2	1061	208416	19003872	816268080	14693508624	81360020352	81360020352	14693508624	816268080	19003872	208416	1061	2		193778020814
14	2	1133	244092	25348908	1316841852	31933874604	296433473916	697026576684	296433473916	31933874604	1316841852	25348908	244092	1133	2	1356446145698

b-files: A109808 (row sums), A153649, A153649 (reversed triangle) ?

Technical details

Node id: n-6db0c6ee051777e9 · revision 0 · JSON
Definition source: oeis_catalog:A153649
Fingerprint: f6-959f5bc62ab0c873

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: yes
γ-positive (gamma_positive): yes
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

A153649 triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?
A153649 reversed triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?

Row sums ? A109808

Submission text ?

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

%e Example

Triangle begins:
  0: 2;
  1: 7,   7;
  2: 2,  94,     2;
  3: 2, 341,   341,       2;
  4: 2, 413,  3972,     413,        2;
  5: 2, 485, 16320,   16320,      485,        2;
  6: 2, 557, 31260,  171660,    31260,      557,       2;
  7: 2, 629, 48792,  774120,   774120,    48792,     629,     2;
  8: 2, 701, 68916, 1917012,  7556340,  1917012,   68916,   701,   2;
  9: 2, 773, 91632, 3693648, 36567552, 36567552, 3693648, 91632, 773, 2;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (n-2+(n-2)*x)*P(n-1,x) + (35*n-70)*x*P(n-2,x) for n >= 3, with P(0,x) = 2, P(1,x) = 7 + 7*x, P(2,x) = 2 + 94*x + 2*x^2.

%t Mathematica

P[0, x_] := 2;
P[1, x_] := 7 + 7*x;
P[2, x_] := 2 + 94*x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 2 + (n - 2)*x)*P[n - 1, x] + (35*n - 70)*x*P[n - 2, x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 7 + 7*x; v[3] = 2 + 94*x + 2*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n - 2 + (n - 2)*x)*v[i - 1] + (35*n - 70)*x*v[i - 2])/(n - 2)); 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

A153649 triangle, offset 1, from P₀

%e Example

Triangle begins:
   1: 2;
   2: 7,   7;
   3: 2,  94,     2;
   4: 2, 341,   341,       2;
   5: 2, 413,  3972,     413,        2;
   6: 2, 485, 16320,   16320,      485,        2;
   7: 2, 557, 31260,  171660,    31260,      557,       2;
   8: 2, 629, 48792,  774120,   774120,    48792,     629,     2;
   9: 2, 701, 68916, 1917012,  7556340,  1917012,   68916,   701,   2;
  10: 2, 773, 91632, 3693648, 36567552, 36567552, 3693648, 91632, 773, 2;

%F Formula

The row polynomials satisfy (n-3)*P(n,x) = (n-3+(n-3)*x)*P(n-1,x) + (35*n-105)*x*P(n-2,x) for n >= 4, with P(1,x) = 2, P(2,x) = 7 + 7*x, P(3,x) = 2 + 94*x + 2*x^2.

%t Mathematica

P[1, x_] := 2;
P[2, x_] := 7 + 7*x;
P[3, x_] := 2 + 94*x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 3 + (n - 3)*x)*P[n - 1, x] + (35*n - 105)*x*P[n - 2, x])/(n - 3)]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 2; v[2] = 7 + 7*x; v[3] = 2 + 94*x + 2*x^2; for(n = 4, N, my(i = n); v[i] = ((n - 3 + (n - 3)*x)*v[i - 1] + (35*n - 105)*x*v[i - 2])/(n - 3)); 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

A153649 reversed triangle, offset 1, from P₀

%e Example

Triangle begins:
   1: 2;
   2: 7,   7;
   3: 2,  94,     2;
   4: 2, 341,   341,       2;
   5: 2, 413,  3972,     413,        2;
   6: 2, 485, 16320,   16320,      485,        2;
   7: 2, 557, 31260,  171660,    31260,      557,       2;
   8: 2, 629, 48792,  774120,   774120,    48792,     629,     2;
   9: 2, 701, 68916, 1917012,  7556340,  1917012,   68916,   701,   2;
  10: 2, 773, 91632, 3693648, 36567552, 36567552, 3693648, 91632, 773, 2;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where (n-3)*P(n,x) = (n-3+(n-3)*x)*P(n-1,x) + (35*n-105)*x*P(n-2,x) for n >= 4, with P(1,x) = 2, P(2,x) = 7 + 7*x, P(3,x) = 2 + 94*x + 2*x^2.

%t Mathematica

P[1, x_] := 2;
P[2, x_] := 7 + 7*x;
P[3, x_] := 2 + 94*x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 3 + (n - 3)*x)*P[n - 1, x] + (35*n - 105)*x*P[n - 2, x])/(n - 3)]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 2; v[2] = 7 + 7*x; v[3] = 2 + 94*x + 2*x^2; for(n = 4, N, my(i = n); v[i] = ((n - 3 + (n - 3)*x)*v[i - 1] + (35*n - 105)*x*v[i - 2])/(n - 3)); 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

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0)
by Liu–Wang theorem for three-term recurrences, Liu–Wang theorem with a lag row
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Liu–Wang theorem for three-term recurrences, Liu–Wang theorem with a lag row
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Liu–Wang theorem for three-term recurrences, Liu–Wang theorem with a lag row
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m ≥ 2
by Liu–Wang theorem with a lag row
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 2
by Liu–Wang theorem for three-term recurrences

Proof chains ? JSON

Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check

References

[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003