A038763: Triangular matrix arising in enumeration of catafusenes, read by rows.

A038763 ↗ nonnegative coefficientsonly rational roots

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

Recurrence from A038763

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

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

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

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	1														2
2	1	4	3													8
3	1	7	15	9												32
4	1	10	36	54	27											128
5	1	13	66	162	189	81										512
6	1	16	105	360	675	648	243									2048
7	1	19	153	675	1755	2673	2187	729								8192
8	1	22	210	1134	3780	7938	10206	7290	2187							32768
9	1	25	276	1764	7182	19278	34020	37908	24057	6561						131072
10	1	28	351	2592	12474	40824	91854	139968	137781	78732	19683					524288
11	1	31	435	3645	20250	78246	214326	415530	557685	492075	255879	59049				2097152
12	1	34	528	4950	31185	138996	449064	1058508	1804275	2165130	1732104	826686	177147			8388608
13	1	37	630	6534	46035	232551	866052	2405700	4979799	7577955	8227494	6022998	2657205	531441		33554432
14	1	40	741	8424	65637	370656	1563705	5003856	12196899	22517352	30961359	30705480	20726199	8503056	1594323	134217728

b-files: A004171 (row sums), A038763, A081294 (row sums), A136158 (reversed triangle) ?

Technical details

Node id: n-685ab5842fbad7f4 · revision 1 · JSON
Definition source: oeis_catalog:A038763
Fingerprint: f6-fb13869215ab1dd7

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): yes

Matches ?

A038763 triangle 66 terms, 11 rows · tabl · nonn · catalog b-file?
A136158 reversed triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?

Row sums ? A004171, A081654, A081294

Submission text ?

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

%e Example

Triangle begins:
  0: 1;
  1: 1,  1;
  2: 1,  4,   3;
  3: 1,  7,  15,    9;
  4: 1, 10,  36,   54,   27;
  5: 1, 13,  66,  162,  189,    81;
  6: 1, 16, 105,  360,  675,   648,   243;
  7: 1, 19, 153,  675, 1755,  2673,  2187,   729;
  8: 1, 22, 210, 1134, 3780,  7938, 10206,  7290,  2187;
  9: 1, 25, 276, 1764, 7182, 19278, 34020, 37908, 24057, 6561;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (n-1+(3*n-3)*x)*P(n-1,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[((n - 1 + (3*n - 3)*x)*P[n - 1, x])/(n - 1)]];
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 + x; for(n = 2, N, my(i = n + 1); v[i] = ((n - 1 + (3*n - 3)*x)*v[i - 1])/(n - 1)); 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

A038763 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0: 1;
  1: 1,  1;
  2: 1,  4,   3;
  3: 1,  7,  15,    9;
  4: 1, 10,  36,   54,   27;
  5: 1, 13,  66,  162,  189,    81;
  6: 1, 16, 105,  360,  675,   648,   243;
  7: 1, 19, 153,  675, 1755,  2673,  2187,   729;
  8: 1, 22, 210, 1134, 3780,  7938, 10206,  7290,  2187;
  9: 1, 25, 276, 1764, 7182, 19278, 34020, 37908, 24057, 6561;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (n-1+(3*n-3)*x)*P(n-1,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[((n - 1 + (3*n - 3)*x)*P[n - 1, 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] = ((n - 1 + (3*n - 3)*x)*v[i - 1])/(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

A136158 reversed triangle, offset 0, from P₀

%e Example

Triangle begins:
  0:    1;
  1:    1,     1;
  2:    3,     4,     1;
  3:    9,    15,     7,     1;
  4:   27,    54,    36,    10,     1;
  5:   81,   189,   162,    66,    13,    1;
  6:  243,   648,   675,   360,   105,   16,    1;
  7:  729,  2187,  2673,  1755,   675,  153,   19,   1;
  8: 2187,  7290, 10206,  7938,  3780, 1134,  210,  22,  1;
  9: 6561, 24057, 37908, 34020, 19278, 7182, 1764, 276, 25, 1;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where (n-1)*P(n,x) = (n-1+(3*n-3)*x)*P(n-1,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[((n - 1 + (3*n - 3)*x)*P[n - 1, 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] = ((n - 1 + (3*n - 3)*x)*v[i - 1])/(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

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), Liu–Wang theorem
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Row-hypergeometric rows (multiplier sequences), Liu–Wang theorem, 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), Liu–Wang theorem, Iterated real-rootedness preservers
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 2
by Liu–Wang theorem
Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Iterated real-rootedness preservers
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 1
by Iterated real-rootedness preservers

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
Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Related ?

Same fingerprint ?

A136158: Triangle whose rows are generated by A136157^n * [1, 1, 0, 0, 0, ...].

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
[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.