A104978: Triangle read by rows, where the g.f. satisfies A(x, y) = 1 + xA(x, y)^2 + xy*A(x, y)^3.

A104978 ↗ nonnegative coefficients

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

Recurrence from A104978

\(\left(2\,n + 4\,n^{2}\right)\,P_{n}(x) = \left(1 + 3\,x - 9\,n - 15\,n x + 14\,n^{2} + 18\,n^{2} x\right)\,P_{n-1}(x) + \left(-1 - 4\,x - 3\,x^{2} + n + 10\,n x + 9\,n x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	1							2
2	2	5	3						10
3	5	21	28	12					66
4	14	84	180	165	55				498
5	42	330	990	1430	1001	273			4066
6	132	1287	5005	10010	10920	6188	1428		34970
7	429	5005	24024	61880	92820	81396	38760	7752	312066

All 15 rows b-files: A027307 (row sums), A104978, A108426 (reversed triangle) ?

Technical details

Node id: n-9ed87ca196739b1c · revision 2 · JSON
Definition source: oeis_catalog:A104978
Fingerprint: f6-da93e8b7035ccffa

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 ?

A108426 reversed triangle 46 terms, 9 rows · tabl · nonn · catalog b-file?
A104978 triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?

Row sums ? A027307

Submission text ?

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

%e Example

Triangle begins:
  0:    1;
  1:    1,     1;
  2:    2,     5,      3;
  3:    5,    21,     28,      12;
  4:   14,    84,    180,     165,      55;
  5:   42,   330,    990,    1430,    1001,     273;
  6:  132,  1287,   5005,   10010,   10920,    6188,    1428;
  7:  429,  5005,  24024,   61880,   92820,   81396,   38760,    7752;
  8: 1430, 19448, 111384,  352716,  678300,  813960,  596904,  245157,   43263;
  9: 4862, 75582, 503880, 1899240, 4476780, 6864396, 6864396, 4326300, 1562275, 246675;

%F Formula

The row polynomials satisfy (4*n^2+2*n)*P(n,x) = (14*n^2-9*n+1+(18*n^2-15*n+3)*x)*P(n-1,x) + (n-1+(10*n-4)*x+(9*n-3)*x^2)*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[((14*n^2 - 9*n + 1 + (18*n^2 - 15*n + 3)*x)*P[n - 1, x] + (n - 1 + (10*n - 4)*x + (9*n - 3)*x^2)*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] = ((14*n^2 - 9*n + 1 + (18*n^2 - 15*n + 3)*x)*v[i - 1] + (n - 1 + (10*n - 4)*x + (9*n - 3)*x^2)*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

A104978 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0:    1;
  1:    1,     1;
  2:    2,     5,      3;
  3:    5,    21,     28,      12;
  4:   14,    84,    180,     165,      55;
  5:   42,   330,    990,    1430,    1001,     273;
  6:  132,  1287,   5005,   10010,   10920,    6188,    1428;
  7:  429,  5005,  24024,   61880,   92820,   81396,   38760,    7752;
  8: 1430, 19448, 111384,  352716,  678300,  813960,  596904,  245157,   43263;
  9: 4862, 75582, 503880, 1899240, 4476780, 6864396, 6864396, 4326300, 1562275, 246675;

%F Formula

The row polynomials satisfy (4*n^2+2*n)*P(n,x) = (14*n^2-9*n+1+(18*n^2-15*n+3)*x)*P(n-1,x) + (n-1+(10*n-4)*x+(9*n-3)*x^2)*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[((14*n^2 - 9*n + 1 + (18*n^2 - 15*n + 3)*x)*P[n - 1, x] + (n - 1 + (10*n - 4)*x + (9*n - 3)*x^2)*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] = ((14*n^2 - 9*n + 1 + (18*n^2 - 15*n + 3)*x)*v[i - 1] + (n - 1 + (10*n - 4)*x + (9*n - 3)*x^2)*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

A108426 reversed triangle, offset 0, from P₀

%e Example

Triangle begins:
  0:      1;
  1:      1,       1;
  2:      3,       5,       2;
  3:     12,      28,      21,       5;
  4:     55,     165,     180,      84,      14;
  5:    273,    1001,    1430,     990,     330,      42;
  6:   1428,    6188,   10920,   10010,    5005,    1287,     132;
  7:   7752,   38760,   81396,   92820,   61880,   24024,    5005,    429;
  8:  43263,  245157,  596904,  813960,  678300,  352716,  111384,  19448,  1430;
  9: 246675, 1562275, 4326300, 6864396, 6864396, 4476780, 1899240, 503880, 75582, 4862;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where (4*n^2+2*n)*P(n,x) = (14*n^2-9*n+1+(18*n^2-15*n+3)*x)*P(n-1,x) + (n-1+(10*n-4)*x+(9*n-3)*x^2)*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[((14*n^2 - 9*n + 1 + (18*n^2 - 15*n + 3)*x)*P[n - 1, x] + (n - 1 + (10*n - 4)*x + (9*n - 3)*x^2)*D[P[n - 1, x], x])/(4*n^2 + 2*n)]];
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; for(n = 1, N, my(i = n + 1); v[i] = ((14*n^2 - 9*n + 1 + (18*n^2 - 15*n + 3)*x)*v[i - 1] + (n - 1 + (10*n - 4)*x + (9*n - 3)*x^2)*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 = Vecrev(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 Affine, reversal and Möbius substitutions (elementary), Iterated real-rootedness preservers
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Row-hypergeometric rows (multiplier sequences), Affine, reversal and Möbius substitutions (elementary), Iterated real-rootedness preservers
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Row-hypergeometric rows (multiplier sequences), Affine, reversal and Möbius substitutions (elementary), Iterated real-rootedness preservers
Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0)
by Row-hypergeometric rows (multiplier sequences)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle read by rows: T(n,k) is number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch th…?
  - 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
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows, where the g.f. satisfies A(x, y) = 1 + x*A(x, y)^2 + x*y*A(x, y)^3.?
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] 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
↑ Row-hypergeometric rows (multiplier sequences) (above)
↑ Iterated real-rootedness preservers (above)

Used in 7

Related ?

Same fingerprint ?

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[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.