A088617: Triangle read by rows: T(n,k) = C(n+k,n)*C(n,k)/(k+1), for n >= 0, k = 0..n.

A088617 ↗ nonnegative coefficients

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

Recurrence from A088617

\(\left(n\right)\,P_{n}(x) = \left(-3 - 6\,x + 2\,n + 4\,n x\right)\,P_{n-1}(x) + \left(3 - n\right)\,P_{n-2}(x)\)

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

\(P_{2}(x) = 1 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	1	1							2
3	1	3	2						6
4	1	6	10	5					22
5	1	10	30	35	14				90
6	1	15	70	140	126	42			394
7	1	21	140	420	630	462	132		1806
8	1	28	252	1050	2310	2772	1716	429	8558

All 15 rows b-files: A006318 (row sums), A060693 (reversed triangle), A088617 ?

Technical details

Node id: n-be392fc9b98fb771 · revision 1 · JSON
Definition source: oeis_catalog:A088617
Fingerprint: f6-3d0fbbfa79b3fd61

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 ?

A060693 reversed triangle 56 terms, 10 rows · tabl · nonn b-file?
A088617 triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?

Row sums ? A155069, A006318, A025240

Submission text ?

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

%e Example

Triangle begins:
   1: 1;
   2: 1,  1;
   3: 1,  3,   2;
   4: 1,  6,  10,    5;
   5: 1, 10,  30,   35,    14;
   6: 1, 15,  70,  140,   126,    42;
   7: 1, 21, 140,  420,   630,   462,   132;
   8: 1, 28, 252, 1050,  2310,  2772,  1716,   429;
   9: 1, 36, 420, 2310,  6930, 12012, 12012,  6435,  1430;
  10: 1, 45, 660, 4620, 18018, 42042, 60060, 51480, 24310, 4862;

%F Formula

The row polynomials satisfy n*P(n,x) = (2*n-3+(4*n-6)*x)*P(n-1,x) - (n-3)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 3 + (4*n - 6)*x)*P[n - 1, x] - (n - 3)*P[n - 2, x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; for(n = 3, N, my(i = n); v[i] = ((2*n - 3 + (4*n - 6)*x)*v[i - 1] - (n - 3)*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, v[i])))};
T(10)

polytool c48629374597

A060693 reversed triangle, offset 0, from P₁

%e Example

Triangle begins:
  0:    1;
  1:    1,     1;
  2:    2,     3,     1;
  3:    5,    10,     6,     1;
  4:   14,    35,    30,    10,     1;
  5:   42,   126,   140,    70,    15,     1;
  6:  132,   462,   630,   420,   140,    21,    1;
  7:  429,  1716,  2772,  2310,  1050,   252,   28,   1;
  8: 1430,  6435, 12012, 12012,  6930,  2310,  420,  36,  1;
  9: 4862, 24310, 51480, 60060, 42042, 18018, 4620, 660, 45, 1;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where (n+1)*P(n,x) = (2*n-1+(4*n-2)*x)*P(n-1,x) - (n-2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 1 + (4*n - 2)*x)*P[n - 1, x] - (n - 2)*P[n - 2, x])/(n + 1)]];
Table[Reverse[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 + x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 1 + (4*n - 2)*x)*v[i - 1] - (n - 2)*v[i - 2])/(n + 1)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

A088617 triangle, offset 0, from P₁

%e Example

Triangle begins:
  0: 1;
  1: 1,  1;
  2: 1,  3,   2;
  3: 1,  6,  10,    5;
  4: 1, 10,  30,   35,    14;
  5: 1, 15,  70,  140,   126,    42;
  6: 1, 21, 140,  420,   630,   462,   132;
  7: 1, 28, 252, 1050,  2310,  2772,  1716,   429;
  8: 1, 36, 420, 2310,  6930, 12012, 12012,  6435,  1430;
  9: 1, 45, 660, 4620, 18018, 42042, 60060, 51480, 24310, 4862;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 1 + (4*n - 2)*x)*P[n - 1, x] - (n - 2)*P[n - 2, x])/(n + 1)]];
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 + x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 1 + (4*n - 2)*x)*v[i - 1] - (n - 2)*v[i - 2])/(n + 1)); 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 ≥ 1, is zero or real-rooted with all roots in (−∞, 0)
by Liu–Wang theorem for three-term recurrences, Row-hypergeometric rows (multiplier sequences)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Liu–Wang theorem for three-term recurrences, Row-hypergeometric rows (multiplier sequences)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Liu–Wang theorem for three-term recurrences, Row-hypergeometric rows (multiplier sequences)
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
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

Related ?

Same fingerprint ?

A253284: Triangle read by rows, T(n,k) = (k+1)*(n+1)!*(n+k)!/((k+1)!^2*(n-k)!) with n >= 0 and 0 <= k <= n.

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.