A112934 -id:A112934 - cp's OEIS frontend

A112937 Logarithmic derivative of A112936 such that a(n)=(1/3)*A112936(n+1) for n>0, where A112936 equals the INVERT transform (with offset) of triple factorials A008544.

1, 5, 37, 377, 4981, 81305, 1580797, 35637377, 913115701, 26189790425, 830916198157, 28883617580177, 1091455878504421, 44541746007215945, 1952125704702209917, 91440056107001450177, 4558596081095404198741

Offset: 1

Paul D. Hanna, Oct 09 2005

nonn

			log(1+x + 3*x*[x + 5*x^2 + 37*x^3 + 377*x^4 + 4981*x^5 +...])
= x + 5/2*x^2 + 37/3*x^3 + 377/4*x^4 + 4981/5*x^5 + ...

Cf. A008544, A112936; A112934, A112935, A112938, A112939, A112940, A112941, A112942, A112943.

{a(n)=local(F=1+x+x*O(x^n));for(i=1,n,F=1+x+3*x^2*deriv(F)/F); return(n*polcoeff(log(F),n,x))}

G.f.: log(1+x + 3*x*[Sum_{k>=1} a(n)]) = Sum_{k>=1} a(n)/n*x^n.

A112939 Logarithmic derivative of A112938 such that a(n)=(1/4)*A112938(n+1) for n>0, where A112938 equals the INVERT transform (with offset) of quadruple factorials A008545.

Original entry on oeis.org

1, 7, 73, 1039, 18961, 423703, 11208793, 342414367, 11855713825, 458600785447, 19594307026537, 916242295851055, 46533732766792753, 2550471781317027127, 150035539128333384313, 9428390893356604340287, 630318228814408172573761

Offset: 1

Paul D. Hanna, Oct 09 2005

nonn

			log(1+x + 4*x*[x + 7*x^2 + 73*x^3 + 1039*x^4 + 18961*x^5 +...])
= x + 7/2*x^2 + 73/3*x^3 + 1039/4*x^4 + 18961/5*x^5 + ...

Cf. A008545, A112938; A112934, A112935, A112936, A112937, A112940, A112941, A112942, A112943.

{a(n)=local(F=1+x+x*O(x^n));for(i=1,n,F=1+x+4*x^2*deriv(F)/F); return(n*polcoeff(log(F),n,x))}

G.f.: log(1+x + 4*x*[Sum_{k>=1} a(n)]) = Sum_{k>=1} a(n)/n*x^n.

G.f.: 1/x - G(0)/(2*x), where G(k)= 1 + 1/(1 - x*(4*k-1)/(x*(4*k-3) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 04 2013

A112941 Logarithmic derivative of A112940 such that a(n)=(1/5)*A112940(n+1) for n>0, where A112940 equals the INVERT transform (with offset) of quintuple factorials A008546.

Original entry on oeis.org

1, 9, 121, 2209, 51401, 1457649, 48774041, 1880312129, 82028211241, 3993290362449, 214543742998201, 12606663551853409, 804145149477634121, 55332318403485181809, 4084986234723143402201, 322064057582671115832449

Offset: 1

Paul D. Hanna, Oct 09 2005

nonn

			log(1+x + 5*x*[x + 9*x^2 + 121*x^3 + 2209*x^4 + 51401*x^5 +...])
= x + 9/2*x^2 + 121/3*x^3 + 2209/4*x^4 + 51401/5*x^5 + ...

Cf. A008546, A112940; A112934, A112935, A112936, A112937, A112938, A112939, A112942, A112943.

{a(n)=local(F=1+x+x*O(x^n));for(i=1,n,F=1+x+5*x^2*deriv(F)/F); return(n*polcoeff(log(F),n,x))}

G.f.: log(1+x + 5*x*[Sum_{n>=1} a(n)]) = Sum_{n>=1} a(n)/n*x^n.

A112943 Logarithmic derivative of A112942 such that a(n)=(1/6)*A112942(n+1) for n>0, where A112942 equals the INVERT transform (with offset) of sextuple factorials A008543.

Original entry on oeis.org

1, 11, 181, 4031, 114001, 3917771, 158531941, 7380184511, 388385146081, 22791211333451, 1475182111403221, 104384110708795391, 8015356365346614961, 663741406196190241931, 58957686544170035607301

Offset: 1

Paul D. Hanna, Oct 09 2005

nonn

			log(1+x + 6*x*[x + 11*x^2 + 181*x^3 + 4031*x^4 + 114001*x^5 +...])
= x + 11/2*x^2 + 181/3*x^3 + 4031/4*x^4 + 114001/5*x^5 + ...

Cf. A008543, A112942; A112934, A112935, A112936, A112937, A112938, A112939, A112940, A112941.

{a(n)=local(F=1+x+x*O(x^n));for(i=1,n,F=1+x+6*x^2*deriv(F)/F); return(n*polcoeff(log(F),n,x))}

G.f.: log(1+x + 6*x*[Sum_{n>=1} a(n)]) = Sum_{n>=1} a(n)/n*x^n.

A113143 Table T(n,k), n >= 0 and k >= 0, read by antidiagonals, related to A111146.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 2, 5, 8, 1, 1, 2, 6, 15, 16, 1, 1, 2, 7, 26, 54, 32, 1, 1, 2, 8, 41, 158, 235, 64, 1, 1, 2, 9, 60, 364, 1282, 1237, 128, 1, 1, 2, 10, 83, 708, 4409, 13158, 7790, 256, 1, 1, 2, 11, 110, 1226, 11428, 67563, 163354

Offset: 0

Philippe Deléham and Paul D. Hanna, Oct 30 2005

Let R(m,n,k), 0 <= k <= n, the Riordan array (1, x*g(x)) where g(x)is g.f. of the m-fold factorials.
Then the row sums of R(m,n,k) are given by row m; example: m = 1, R(1,n,k) = A084938(n,k) and A051295 gives the row sums of A084938.
Square array of INVERT of m-fold factorials.

			Table begins:
1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, ...
1, 1, 2, 5, 15, 54, 235, 1237, 7790, 57581, 489231, 4690254, ...
1, 1, 2, 6, 26, 158, 1282, 13158, 163354, 2374078, 39456386, ...
1, 1, 2, 7, 41, 364, 4409, 67573, 1248626, 26948347, 664414997, ...
1, 1, 2, 8, 60, 708, 11428, 232756, 5704964, 163192820, 5331728964, ...
1, 1, 2, 9, 83, 1226, 24727, 627909, 19169758, 682800001, 27776711627, ...
1, 1, 2, 10, 110, 1954, 47270, 1437562, 52531310, 2239259266, 109021857446, ...
1, 1, 2, 11, 141, 2928, 82597, 2925973, 124502114, 6179425823, 350316271761, ...
1, 1, 2, 12, 176, 4184, 134824, 5451528, 264710536, 14992543432, 969925065992, ...
1, 1, 2, 13, 215, 5758, 208643, 9481141, 517310894, 32922122485, 2393313188039, ...
1, 1, 2, 14, 258, 7686, 309322, 15604654, 945111938, 66766075046, 5387893860042, ...

Cf. A051295 (row n=1), A112934 (row n=2), A113144 (row n = 3), A113145 (row n=4), A113146 (row n=5), A113147 (row n = 6), A113148 (row n=7), A113149 (row n=8).

{T(n,k)=local(x=X+X*O(X^k),y=Y+Y*O(Y^k));A=1/(1-x*y*sum(j=0,k,x^j*prod(i= 0,j-1,y+i)));return(sum(m=0,k,n^(k-m)*polcoeff(polcoeff(A,k,X),m,Y)))}

T(n, k) = Sum_{j=0..k} n^(k-j)*A111146(k, j).

A365673 Array A(n, k) read by ascending antidiagonals. Polygonal number weighted generalized Catalan sequences.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 4, 1, 1, 1, 4, 15, 8, 1, 1, 1, 5, 34, 105, 16, 1, 1, 1, 6, 61, 496, 945, 32, 1, 1, 1, 7, 96, 1385, 11056, 10395, 64, 1, 1, 1, 8, 139, 2976, 50521, 349504, 135135, 128, 1, 1, 1, 9, 190, 5473, 151416, 2702765, 14873104, 2027025, 256, 1

Offset: 0

Peter Luschny, Sep 30 2023

Using polygonal numbers as weights, a recursion for triangles is defined, whose main diagonals represents a family of sequences, which include, among others, the powers of 2, the double factorial of odd numbers, the reduced tangent numbers, and the Euler numbers.
Apart from the edge cases k = 0 and k = n the recursion is T(n, k) = w(n, k) * T(n, k - 1) + T(n - 1, k). T(n, 0) = 1 and T(n, n) = T(n, n-1) if n > 0.
The weights w(n, k) identical to 1 yield the recursion of the Catalan triangle A009766 (with main diagonal the Catalan numbers). Here the polygonal numbers are used as weights in the form w(n, k) = p(s, n - k + 1), where the parameter s is the number of sides of the polygon and p(s, n) = ((s-2) * n^2 - (s-4) * n) / 2, see A317302.

			Array A(n, k) starts:                            (polygon|diagonal|triangle)
[0] 1, 1, 1,   1,     1,       1,         1, ...  A258837  A000012
[1] 1, 1, 2,   4,     8,      16,        32, ...  A080956  A011782
[2] 1, 1, 3,  15,   105,     945,     10395, ...  A001477  A001147  A001498
[3] 1, 1, 4,  34,   496,   11056,    349504, ...  A000217  A002105  A365674
[4] 1, 1, 5,  61,  1385,   50521,   2702765, ...  A000290  A000364  A060058
[5] 1, 1, 6,  96,  2976,  151416,  11449296, ...  A000326  A126151  A366138
[6] 1, 1, 7, 139,  5473,  357721,  34988647, ...  A000384  A126156  A365672
[7] 1, 1, 8, 190,  9080,  725320,  87067520, ...  A000566  A366150  A366149
[8] 1, 1, 9, 249, 14001, 1322001, 188106489, ...  A000567
           A054556                         A366137

Wikipedia, Polygonal number.

Poly weights: A258837, A080956, A001477, A000217, A000290, A000326, A000384.
Rows: A000012, A011782, A001147, A002105, A000364, A126151, A126156, A366150.
Triangles: A001498, A365674, A060058, A366138, A365672, A366149.
Cf. A009766, A366137 (central diagonal), A317302 (table of polygonal numbers).
Cf. A112934, A303943, A305532, A305533.

poly := (s, n) -> ((s - 2) * n^2 - (s - 4) * n) / 2:
T := proc(s, n, k) option remember; if k = 0 then 1 else if k = n then T(s, n, k-1) else poly(s, n - k + 1) * T(s, n, k - 1) + T(s, n - 1, k) fi fi end:
for n from 0 to 8 do A := (n, k) -> T(n, k, k): seq(A(n, k), k = 0..9) od;
# Alternative, using continued fractions:
A := proc(p, L) local CF, poly, k, m, P, ser;
   poly := (s, n) -> ((s - 2)*n^2 - (s - 4)*n)/2;
   CF := 1 + x;
   for k from 1 to L do
       m := L - k + 1;
       P := poly(p, m);
       CF := 1/(1 - P*x*CF)
   od;
   ser := series(CF, x, L);
   seq(coeff(ser, x, m), m = 0..L-1)
end:
for p from 0 to 8 do lprint(A(p, 8)) od;

poly[s_, n_] := ((s - 2) * n^2 - (s - 4) * n) / 2;
T[s_, n_, k_] := T[s, n, k] = If[k == 0, 1, If[k == n, T[s, n, k - 1], poly[s, n - k + 1] * T[s, n, k - 1] + T[s, n - 1, k]]];
A[n_, k_] := T[n, k, k];
Table[A[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 27 2023, from first Maple program *)

A(p, n) = {
       my(CF = 1 + x,
           poly(s, n) = ((s - 2)*n^2 - (s - 4)*n)/2,
           m, P
       );
       for(k = 1, n,
           m = n - k + 1;
           P = poly(p, m);
           CF = 1/(1 - P*x*CF)
        );
        Vec(CF + O(x^(n)))
}
for(p = 0, 8, print(A(p, 8)))
\\  Michel Marcus and Peter Luschny, Oct 02 2023

from functools import cache
@cache
def T(s, n, k):
    if k == 0: return 1
    if k == n: return T(s, n, k - 1)
    p = (n - k + 1) * ((s - 2) * (n - k + 1) - (s - 4)) // 2
    return p * T(s, n, k - 1) + T(s, n - 1, k)
def A(n, k): return T(n, k, k)
for n in range(9): print([A(n, k) for k in range(9)])

A111106 Riordan array (1, x*g(x)) where g(x) is g.f. of double factorials A001147.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 3, 2, 1, 0, 15, 7, 3, 1, 0, 105, 36, 12, 4, 1, 0, 945, 249, 64, 18, 5, 1, 0, 10395, 2190, 441, 100, 25, 6, 1, 0, 135135, 23535, 3807, 691, 145, 33, 7, 1, 0, 2027025, 299880, 40032, 5880, 1010, 200, 42, 8, 1

Offset: 0

Philippe Deléham, Oct 13 2005, Dec 20 2008

Triangle T(n,k), 0 <= k <= n, given by [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

			Rows begin:
  1;
  0,       1;
  0,       1,      1;
  0,       3,      2,     1;
  0,      15,      7,     3,    1;
  0,     105,     36,    12,    4,    1;
  0,     945,    249,    64,   18,    5,   1;
  0,   10395,   2190,   441,  100,   25,   6,  1:
  0,  135135,  23535,  3807,  691,  145,  33,  7, 1;
  0, 2027025, 299880, 40032, 5880, 1010, 200, 42, 8, 1;

Columns: A000007, A001147, A034430; diagonals: A000012, A001477, A055998.

Cf. A084938, A111088, A112934, A168441.

# Uses function PMatrix from A357368.
PMatrix(10, n -> doublefactorial(2*n-3)); # Peter Luschny, Oct 19 2022

T(n, k) = Sum_{j=0..n-k} T(n-1, k-1+j)*A111088(j).
Sum_{k=0..n} T(n, k) = A112934(n).
G.f.: 1/(1-xy/(1-x/(1-2x/(1-3x/(1-4x/(1-... (continued fraction). - Paul Barry, Jan 29 2009
Sum_{k=0..n} T(n,k)*2^(n-k) = A168441(n). - Philippe Deléham, Nov 28 2009

A355722 Row 2 of table A355721.

Original entry on oeis.org

1, 2, 14, 138, 1686, 24162, 394254, 7191018, 144786006, 3188449602, 76246683534, 1968284351178, 54576250392726, 1618348891438242, 51122453577462414, 1714406473587300138, 60843580566100937046, 2278637898592632599682, 89818339421620249242894, 3717488491001699691500298

Offset: 0

Peter Bala, Jul 15 2022

A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.

Cf. A001147, A355721 (table), A112934 (row 0), A000698 (row 1), A355723 (row 3), A355724 (row 4), A355725 (row 5).

n := 2: seq(coeff(series( hypergeom([n+1/2, 1], [], 2*x)/hypergeom([n-1/2, 1], [], 2*x ), x, 21), x, k), k = 0..20);

O.g.f: A(x) = ( Sum_{k >= 0} d(k+2)/d(2)*x^k )/( Sum_{k >= 0} d(k+1)/d(1)*x^k ), where d(n) = Product_{k = 1..n} (2*k-1) = A001147(n).
A(x)= 1/(1 + 3*x - 5*x/(1 + 5*x - 7*x/(1 + 7*x - 9*x/(1 + 9*x - ... )))).
The o.g.f. satisfies the Riccati differential equation 2*x^2*A'(x) + 3*x*A(x)^2 - (1 + x)*A(x) + 1 = 0 with A(0) = 1.
Applying Stokes 1982 gives A(x) = 1/(1 - 2*x/(1 - 5*x/(1 - 4*x/(1 - 7*x/(1 - 6*x/(1 - 9*x/(1 - ... - 2*n*x/(1 - (2*n+3)*x )))))))), a continued fraction of Stieltjes type.

A355723 Row 3 of table A355721.

Original entry on oeis.org

1, 2, 18, 218, 3194, 53890, 1019250, 21256090, 483426010, 11895873410, 314834663250, 8918883839450, 269367643864250, 8643467766472450, 293770652998691250, 10546424484691428250, 398914704362503668250, 15860639479547463637250, 661439858772303085871250, 28874834455755565593004250

Offset: 0

Peter Bala, Jul 15 2022

A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.

Cf. A001147, A355721 (table), A112934 (row 0), A000698 (row 1), A355722 (row 2), A355724 (row 4), A355725 (row 5).

n := 3: seq(coeff(series( hypergeom([n+1/2, 1], [], 2*x)/hypergeom([n-1/2, 1], [], 2*x ), x, 21), x, k), k = 0..20);

O.g.f: A(x) = ( Sum_{k >= 0} d(k+3)/d(3)*x^k )/( Sum_{k >= 0} d(k+2)/d(2)*x^k ), where d(n) = Product_{k = 1..n} (2*k-1) = A001147(n).
A(x) = 1/(1 + 5*x - 7*x/(1 + 7*x - 9*x/(1 + 9*x - 11*x/(1 + 11*x - ... )))).
The o.g.f. satisfies the Riccati differential equation 2*x^2*A'(x) + 5*x*A(x)^2 - (1 + 3*x)*A(x) + 1 = 0 with A(0) = 1.
Applying Stokes 1982 gives A(x) = 1/(1 - 2*x/(1 - 7*x/(1 - 4*x/(1 - 9*x/(1 - 6*x/(1 - 11*x/(1 - ... - 2*n*x/(1 - (2*n+5)*x )))))))), a continued fraction of Stieltjes type.

A355724 Row 4 of table A355721.

Original entry on oeis.org

1, 2, 22, 314, 5326, 102722, 2197558, 51355514, 1297759918, 35208930050, 1020115715542, 31432396066106, 1026506419425550, 35428218801977666, 1288967076156307702, 49323199246104202874, 1980947315202528449518, 83342865788161594337282, 3666525676611059535630742

Offset: 0

Peter Bala, Jul 15 2022

A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.

Cf. A001147, A355721 (table), A112934 (row 0), A000698 (row 1), A355722 (row 2), A355723 (row 3), A355725 (row 5).

n := 4: seq(coeff(series( hypergeom([n+1/2, 1], [], 2*x)/hypergeom([n-1/2, 1], [], 2*x ), x, 21), x, k), k = 0..20);

O.g.f: A(x) = ( Sum_{k >= 0} d(k+4)/d(4)*x^k )/( Sum_{k >= 0} d(k+3)/d(3)*x^k ), where d(n) = Product_{k = 1..n} (2*k-1) = A001147(n).
A(x) = 1/(1 + 7*x - 9*x/(1 + 9*x - 11*x/(1 + 11*x - 13*x/(1 + 13*x - ... )))).
The o.g.f. satisfies the Riccati differential equation 2*x^2*A'(x) + 7*x*A(x)^2 - (1 + 5*x)*A(x) + 1 = 0 with A(0) = 1.
Applying Stokes 1982 gives A(x) = 1/(1 - 2*x/(1 - 9*x/(1 - 4*x/(1 - 11*x/(1 - 6*x/(1 - 13*x/(1 - ... - 2*n*x/(1 - (2*n+7)*x )))))))), a continued fraction of Stieltjes-type.

cp's OEIS Frontend

A112937 Logarithmic derivative of A112936 such that a(n)=(1/3)*A112936(n+1) for n>0, where A112936 equals the INVERT transform (with offset) of triple factorials A008544.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula

A112939 Logarithmic derivative of A112938 such that a(n)=(1/4)*A112938(n+1) for n>0, where A112938 equals the INVERT transform (with offset) of quadruple factorials A008545.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula

A112941 Logarithmic derivative of A112940 such that a(n)=(1/5)*A112940(n+1) for n>0, where A112940 equals the INVERT transform (with offset) of quintuple factorials A008546.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula

A112943 Logarithmic derivative of A112942 such that a(n)=(1/6)*A112942(n+1) for n>0, where A112942 equals the INVERT transform (with offset) of sextuple factorials A008543.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula

A113143 Table T(n,k), n >= 0 and k >= 0, read by antidiagonals, related to A111146.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

A365673 Array A(n, k) read by ascending antidiagonals. Polygonal number weighted generalized Catalan sequences.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

A111106 Riordan array (1, x*g(x)) where g(x) is g.f. of double factorials A001147.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Formula

A355722 Row 2 of table A355721.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Formula

A355723 Row 3 of table A355721.