Real-rooted atlas

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

A104978 ↗ nonnegative coefficients

Proved for all n all roots real, in (−∞, 0)coefficients of one signHadamard 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 ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1112
225310
3521281266
4148418016555498
542330990143010012734066
61321287500510010109206188142834970
7429500524024618809282081396387607752312066
8143019448111384352716678300813960596904245157432632862562
948627558250388018992404476780686439668643964326300156227524667526824386
1016796293930223839098062802745758451482970656155505624190031081050100150051430715255680170
115878611440669806280490314001593520503543239705511706206009003004506752252217608256451224084146402471150402
122080124457400424938802390280758858099252283421140420630210055883727905322259800354817320015735376804172259005006710824161357010
137429001738386018253053011417105704757127375139709319752980465488046835886240542870499604589484900027536909400111148979762707475148300830572238552980386
1426744406786391577843902553625799502483721240081962800920199052516520361913666400495664369200509432823900387168946164211183061544782159487201762007636018227665202375085745978

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

Proof chains ? JSON

Used in 7

Same fingerprint ?

References

  1. [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
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.