A357613: Triangle read by rows T(n, k) = binomial(2n, k) binomial(3n - k, 2n).

A357613 ↗ nonnegative coefficients

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

Recurrence from A357613

\(\left(-2\,n + 4\,n^{2}\right)\,P_{n}(x) = \left(6 + 6\,x - 27\,n - 26\,n x - n x^{2} + 27\,n^{2} + 24\,n^{2} x + n^{2} x^{2}\right)\,P_{n-1}(x) + \left(3\,x + 3\,x^{2} - 9\,n x - 10\,n x^{2} - n x^{3}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	3	2							5
2	15	20	6						41
3	84	168	105	20					377
4	495	1320	1260	504	70				3649
5	3003	10010	12870	7920	2310	252			36365
6	18564	74256	120120	100100	45045	10296	924		369305
7	116280	542640	1058148	1113840	680680	240240	45045	3432	3800305

All 15 rows b-files: A026000 (row sums), A357613 ?

Technical details

Node id: n-1397bdceeefc7b3d · revision 0 · JSON
Definition source: oeis_catalog:A357613
Fingerprint: f6-1cc143b67a71a4b2

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 ?

A357613 triangle 36 terms, 8 rows · tabl · nonn · catalog b-file?

Row sums ? A026000

Submission text ?

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

%e Example

Triangle begins:
  0:       1;
  1:       3,        2;
  2:      15,       20,        6;
  3:      84,      168,      105,        20;
  4:     495,     1320,     1260,       504,        70;
  5:    3003,    10010,    12870,      7920,      2310,      252;
  6:   18564,    74256,   120120,    100100,     45045,    10296,      924;
  7:  116280,   542640,  1058148,   1113840,    680680,   240240,    45045,    3432;
  8:  735471,  3922512,  8953560,  11395440,   8817900,  4232592,  1225224,  194480,  12870;
  9: 4686825, 28120950, 73547100, 109830336, 102965940, 62674920, 24690120, 6046560, 831402, 48620;

%F Formula

The row polynomials satisfy (4*n^2-2*n)*P(n,x) = (27*n^2-27*n+6+(24*n^2-26*n+6)*x+(n^2-n)*x^2)*P(n-1,x) - ((9*n-3)*x+(10*n-3)*x^2+n*x^3)*d/dx 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[Together[((27*n^2 - 27*n + 6 + (24*n^2 - 26*n + 6)*x + (n^2 - n)*x^2)*P[n - 1, x] - ((9*n - 3)*x + (10*n - 3)*x^2 + n*x^3)*D[P[n - 1, x], x])/(4*n^2 - 2*n)]];
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] = ((27*n^2 - 27*n + 6 + (24*n^2 - 26*n + 6)*x + (n^2 - n)*x^2)*v[i - 1] - ((9*n - 3)*x + (10*n - 3)*x^2 + n*x^3)*deriv(v[i - 1], x))/(4*n^2 - 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

A357613 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0:       1;
  1:       3,        2;
  2:      15,       20,        6;
  3:      84,      168,      105,        20;
  4:     495,     1320,     1260,       504,        70;
  5:    3003,    10010,    12870,      7920,      2310,      252;
  6:   18564,    74256,   120120,    100100,     45045,    10296,      924;
  7:  116280,   542640,  1058148,   1113840,    680680,   240240,    45045,    3432;
  8:  735471,  3922512,  8953560,  11395440,   8817900,  4232592,  1225224,  194480,  12870;
  9: 4686825, 28120950, 73547100, 109830336, 102965940, 62674920, 24690120, 6046560, 831402, 48620;

%F Formula

The row polynomials satisfy (4*n^2-2*n)*P(n,x) = (27*n^2-27*n+6+(24*n^2-26*n+6)*x+(n^2-n)*x^2)*P(n-1,x) - ((9*n-3)*x+(10*n-3)*x^2+n*x^3)*d/dx 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[Together[((27*n^2 - 27*n + 6 + (24*n^2 - 26*n + 6)*x + (n^2 - n)*x^2)*P[n - 1, x] - ((9*n - 3)*x + (10*n - 3)*x^2 + n*x^3)*D[P[n - 1, x], x])/(4*n^2 - 2*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; for(n = 1, N, my(i = n + 1); v[i] = ((27*n^2 - 27*n + 6 + (24*n^2 - 26*n + 6)*x + (n^2 - n)*x^2)*v[i - 1] - ((9*n - 3)*x + (10*n - 3)*x^2 + n*x^3)*deriv(v[i - 1], x))/(4*n^2 - 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

Used in 7

Related ?

Same fingerprint ?

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.