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.

A253284 ↗ nonnegative coefficients

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

Recurrence from A253284

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	2														4
2	6	18	12													36
3	24	144	240	120												528
4	120	1200	3600	4200	1680											10800
5	720	10800	50400	100800	90720	30240										283680
6	5040	105840	705600	2116800	3175200	2328480	665280									9102240
7	40320	1128960	10160640	42336000	93139200	111767040	69189120	17297280								345058560
8	362880	13063680	152409600	838252800	2514758400	4358914560	4358914560	2335132800	518918400							15090727680
9	3628800	163296000	2395008000	16765056000	65383718400	152562009600	217945728000	186810624000	88216128000	17643225600						747888422400
10	39916800	2195424000	39517632000	342486144000	1678182105600	5034546316800	9589612032000	11644528896000	8733396672000	3687434150400	670442572800					41422381862400
11	479001600	31614105600	684972288000	7192209024000	43153254144000	161105482137600	391256170905600	628804560384000	663738147072000	442492098048000	168951528345600	28158588057600				2535569103513600
12	6227020800	485707622400	12466495641600	155831195520000	1121984607744000	5086330221772800	15258990665318400	31062945282969600	43142979559680000	40266780922368000	24160068553420800	8419417829222400	1295295050649600			169983582318950400
13	87178291200	7933224499200	237996734976000	3490618779648000	29670259627008000	160219401985843200	579841645282099200	1449604113205248000	2536807198109184000	3100542131022336000	2593180691400499200	1414462195309363200	453353267727360000	64764752532480000		12386182292118835200
14	1307674368000	137305808640000	4759934699520000	80918889891840000	801097009929216000	5073614396218368000	21744061698078720000	65232185094236160000	139524395896005120000	213937407040541184000	233386262226044928000	176807774413670400000	88403887206835200000	26229724775654400000	3497296636753920000	974723523832041984000

b-files: A253284 ?

Technical details

Node id: n-694017a14c5c1f2e · revision 1 · JSON
Definition source: oeis_catalog:A253284
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 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 ?

A253284 triangle 36 terms, 8 rows · tabl · nonn · catalog b-file?

Row sums ? A052716

Submission text ?

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

%e Example

Triangle begins:
  0:       1;
  1:       2,         2;
  2:       6,        18,         12;
  3:      24,       144,        240,         120;
  4:     120,      1200,       3600,        4200,        1680;
  5:     720,     10800,      50400,      100800,       90720,        30240;
  6:    5040,    105840,     705600,     2116800,     3175200,      2328480,       665280;
  7:   40320,   1128960,   10160640,    42336000,    93139200,    111767040,     69189120,     17297280;
  8:  362880,  13063680,  152409600,   838252800,  2514758400,   4358914560,   4358914560,   2335132800,   518918400;
  9: 3628800, 163296000, 2395008000, 16765056000, 65383718400, 152562009600, 217945728000, 186810624000, 88216128000, 17643225600;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[(2*n - 1 + (4*n - 2)*x)*P[n - 1, x] - (n^2 - 2*n)*P[n - 2, x]];
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] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (2*n - 1 + (4*n - 2)*x)*v[i - 1] - (n^2 - 2*n)*v[i - 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

A253284 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0:       1;
  1:       2,         2;
  2:       6,        18,         12;
  3:      24,       144,        240,         120;
  4:     120,      1200,       3600,        4200,        1680;
  5:     720,     10800,      50400,      100800,       90720,        30240;
  6:    5040,    105840,     705600,     2116800,     3175200,      2328480,       665280;
  7:   40320,   1128960,   10160640,    42336000,    93139200,    111767040,     69189120,     17297280;
  8:  362880,  13063680,  152409600,   838252800,  2514758400,   4358914560,   4358914560,   2335132800,   518918400;
  9: 3628800, 163296000, 2395008000, 16765056000, 65383718400, 152562009600, 217945728000, 186810624000, 88216128000, 17643225600;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[(2*n - 1 + (4*n - 2)*x)*P[n - 1, x] - (n^2 - 2*n)*P[n - 2, x]];
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] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (2*n - 1 + (4*n - 2)*x)*v[i - 1] - (n^2 - 2*n)*v[i - 2]); 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 Liu–Wang theorem for three-term recurrences, Row-hypergeometric rows (multiplier sequences)
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, Row-hypergeometric rows (multiplier sequences)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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 ≥ 1
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 ?

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