A121207
Triangle read by rows. The definition is by diagonals. The r-th diagonal from the right, for r >= 0, is given by b(0) = b(1) = 1; b(n+1) = Sum_{k=0..n} binomial(n+2,k+r)*a(k).
Original entry on oeis.org
1, 1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 9, 15, 1, 1, 5, 14, 31, 52, 1, 1, 6, 20, 54, 121, 203, 1, 1, 7, 27, 85, 233, 523, 877, 1, 1, 8, 35, 125, 400, 1101, 2469, 4140, 1, 1, 9, 44, 175, 635, 2046, 5625, 12611, 21147, 1, 1, 10, 54, 236, 952, 3488, 11226, 30846, 69161, 115975
Offset: 0
Triangle begins (compare also table 9.2 in the Gould-Quaintance reference):
1;
1, 1;
1, 1, 2;
1, 1, 3, 5;
1, 1, 4, 9, 15;
1, 1, 5, 14, 31, 52;
1, 1, 6, 20, 54, 121, 203;
1, 1, 7, 27, 85, 233, 523, 877;
1, 1, 8, 35, 125, 400,1101, 2469, 4140;
1, 1, 9, 44, 175, 635,2046, 5625, 12611, 21147;
1, 1, 10, 54, 236, 952,3488,11226, 30846, 69161, 115975;
1, 1, 11, 65, 309,1366,5579,20425, 65676,180474, 404663, 678570;
1, 1, 12, 77, 395,1893,8494,34685,126817,407787,1120666,2512769,4213597;
- Alois P. Heinz, Rows n = 0..140, flattened
- Robert Dougherty-Bliss, Gosper's algorithm and Bell numbers, arXiv:2210.13520 [cs.SC], 2022.
- Robert Dougherty-Bliss, Experimental Methods in Number Theory and Combinatorics, Ph. D. Dissertation, Rutgers Univ. (2024). See pp. 69-70.
- H. W. Gould and Jocelyn Quaintance, A linear binomial recurrence and the Bell numbers and polynomials, Applicable Analysis and Discrete Mathematics, 1 (2007), 371-385.
-
function Gould_diag(diag, size)
size < 1 && return []
size == 1 && return [1]
L = [1, 1]
accu = ones(BigInt, diag)
for _ in 1:size-2
accu = cumsum(vcat(accu[end], accu))
L = vcat(L, accu[end])
end
L end # Peter Luschny, Mar 30 2022
-
# This is the Jovovic formula with general index 'd'
# where A040027, A045499, etc. use one explicit integer
# Index n+1 is shifted to n from the original formula.
Gould := proc(n, d) local k;
if n <= 1 then return 1 else
return add(binomial(n-1+d, k+d)*Gould(k, d), k=0..n-1);
fi
end:
# row and col refer to the extrapolated super-table:
# working up to row, not row-1, shows also the Bell numbers
# at the end of each row.
for row from 0 to 13 do
for col from 0 to row do
# 'diag' is constant for one of A040027, A045499 etc.
diag := row - col;
printf("%4d, ", Gould(col, diag));
od;
print();
od; # R. J. Mathar
# second Maple program:
T:= proc(n, k) option remember; `if`(k=0, 1,
add(T(n-j, k-j)*binomial(n-1, j-1), j=1..k))
end:
seq(seq(T(n, k), k=0..n), n=0..12); # Alois P. Heinz, Jan 08 2018
-
g[n_ /; n <= 1, ] := 1; g[n, d_] := g[n, d] = Sum[ Binomial[n-1+d, k+d]*g[k, d], {k, 0, n-1}]; Flatten[ Table[ diag = row-col; g[col, diag], {row, 0, 13}, {col, 0, row}]] (* Jean-François Alcover, Nov 25 2011, after R. J. Mathar *)
T[n_, k_] := T[n, k] = If[k == 0, 1, Sum[T[n-j, k-j] Binomial[n-1, j-1], {j, 1, k}]]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 26 2018, after Alois P. Heinz *)
-
# Computes the n-th diagonal of the triangle reading from the right.
from itertools import accumulate
def Gould_diag(diag, size):
if size < 1: return []
if size == 1: return [1]
L, accu = [1,1], [1]*diag
for _ in range(size-2):
accu = list(accumulate([accu[-1]] + accu))
L.append(accu[-1])
return L # Peter Luschny, Apr 24 2016
A124496
Triangle read by rows: T(n,k) is the number of set partitions of {1,2,...,n} in which the size of the last block is k, 1<=k<=n; the blocks are ordered with increasing least elements.
Original entry on oeis.org
1, 1, 1, 3, 1, 1, 9, 4, 1, 1, 31, 14, 5, 1, 1, 121, 54, 20, 6, 1, 1, 523, 233, 85, 27, 7, 1, 1, 2469, 1101, 400, 125, 35, 8, 1, 1, 12611, 5625, 2046, 635, 175, 44, 9, 1, 1, 69161, 30846, 11226, 3488, 952, 236, 54, 10, 1, 1, 404663, 180474, 65676, 20425, 5579, 1366, 309, 65, 11, 1, 1
Offset: 1
T(4,2) = 4 because we have 13|24, 14|23, 12|34 and 1|2|34.
Triangle starts:
1;
1,1;
3,1,1;
9,4,1,1;
31,14,5,1,1;
121,54,20,6,1,1;
523,233,85,27,7,1,1;
2469,1101,400,125,35,8,1,1;
12611,5625,2046,635,175,44,9,1,1;
69161,30846,11226,3488,952,236,54,10,1,1;
404663,180474,65676,20425,5579,1366,309,65,11,1,1;
2512769,1120666,407787,126817,34685,8494,1893,395,77,12,1,1;
...
-
Q[1]:=t*s: for n from 2 to 12 do Q[n]:=expand(t*s*subs(t=1,Q[n-1])+s*diff(Q[n-1],s)+t*Q[n-1]-Q[n-1]) od:for n from 1 to 12 do P[n]:=sort(subs(s=1,Q[n])) od: for n from 1 to 12 do seq(coeff(P[n],t,j),j=1..n) od;
# second Maple program:
T:= proc(n, k) option remember; `if`(n=k, 1,
add(T(n-j, k)*binomial(n-1, j-1), j=1..n-k))
end:
seq(seq(T(n, k), k=1..n), n=1..12); # Alois P. Heinz, Jul 05 2016
-
T[n_, k_] := T[n, k] = If[n == k, 1, Sum[T[n-j, k]*Binomial[n-1, j-1], {j, 1, n-k}]];
Table[Table[T[n, k], {k, 1, n}], {n, 1, 12}] // Flatten; (* Jean-François Alcover, Jul 21 2016, after Alois P. Heinz *)
A045500
Fifth-from-right diagonal of triangle A121207.
Original entry on oeis.org
1, 1, 6, 27, 125, 635, 3488, 20425, 126817, 831915, 5744784, 41618459, 315388311, 2493721645, 20526285716, 175529425815, 1556577220651, 14290644428279, 135624265589086, 1328702240382589, 13420603191219111, 139592874355534071
Offset: 0
- See also references under sequence A040027.
-
a[0] = a[1] = 1; a[n_] := a[n] = Sum[Binomial[n+3, k+4]*a[k], {k, 0, n-1}];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Jul 14 2018, after Vladeta Jovovic *)
-
{a(n)=local(A=1+x); for(i=1, n, A=1+x*subst(A, x, x/(1-x+x*O(x^n)))/(1-x)^5); polcoeff(A, n)} /* Paul D. Hanna, Mar 23 2012 */
-
# The function Gould_diag is defined in A121207.
A045500_list = lambda size: Gould_diag(5, size)
print(A045500_list(24)) # Peter Luschny, Apr 24 2016
A045501
Third-from-right diagonal of triangle A121207.
Original entry on oeis.org
1, 1, 4, 14, 54, 233, 1101, 5625, 30846, 180474, 1120666, 7352471, 50772653, 367819093, 2787354668, 22039186530, 181408823710, 1551307538185, 13756835638385, 126298933271289, 1198630386463990, 11742905240821910
Offset: 1
-
a[1] = a[2] = 1; a[n_] := a[n] = Sum[Binomial[n, k+1]*a[k], {k, 0, n-1}];
Array[a, 22] (* Jean-François Alcover, Jul 14 2018, after Vladeta Jovovic *)
-
{a(n)=local(A=x+x^2); for(i=1, n, A=x+x*subst(A, x, x/(1-x+x*O(x^n)))/(1-x)^2); polcoeff(A, n)} /* Paul D. Hanna, Mar 23 2012 */
-
# The function Gould_diag is defined in A121207.
A045501_list = lambda size: Gould_diag(3, size)
print(A045501_list(24)) # Peter Luschny, Apr 24 2016
A351817
G.f. A(x) satisfies: A(x) = 1 + x * A(x/(1 - x)^4) / (1 - x)^4.
Original entry on oeis.org
1, 1, 5, 23, 139, 1052, 9166, 90073, 989205, 11981051, 158149438, 2255926638, 34549223880, 564898101239, 9812669832553, 180324597042263, 3492960489714519, 71092066388237562, 1516044005669227542, 33788707128788508476, 785270646437483414261, 18992014442689191510460
Offset: 0
-
nmax = 21; A[] = 0; Do[A[x] = 1 + x A[x/(1 - x)^4]/(1 - x)^4 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n + 3 k + 2, n - k - 1] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 21}]
A346059
G.f. A(x) satisfies: A(x) = 1 - x * A(x/(1 - x)) / (1 - x)^4.
Original entry on oeis.org
1, -1, -3, -2, 15, 62, 56, -566, -3318, -6241, 33022, 330939, 1211873, -1330691, -47459905, -310788796, -675462411, 7151217040, 93213242926, 515144576280, 122725585740, -27551616750331, -296570472858772, -1477869678576483, 3416889475636695, 146832017085068163, 1522825949942199537
Offset: 0
-
nmax = 26; A[] = 0; Do[A[x] = 1 - x A[x/(1 - x)]/(1 - x)^4 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = -Sum[Binomial[n + 2, k + 3] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 26}]
A351283
G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 - x)) / (1 - x)^4.
Original entry on oeis.org
1, 1, 1, 5, 16, 46, 142, 496, 1888, 7538, 31291, 135739, 617461, 2939215, 14575027, 75014471, 399901294, 2205630124, 12572140372, 73961880118, 448447331338, 2798640572516, 17956583819425, 118336081817953, 800278211629795, 5549154792085813, 39420390891260821
Offset: 0
-
nmax = 26; A[] = 0; Do[A[x] = 1 + x + x^2 A[x/(1 - x)]/(1 - x)^4 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = a[1] = 1; a[n_] := a[n] = Sum[Binomial[n + 1, k + 3] a[k], {k, 0, n - 2}]; Table[a[n], {n, 0, 26}]
A352861
a(n) = 1 + Sum_{k=0..n-1} binomial(n+2,k+3) * a(k).
Original entry on oeis.org
1, 2, 7, 28, 121, 570, 2911, 15968, 93433, 580162, 3806275, 26284368, 190415809, 1442982350, 11409436363, 93913277608, 803094241309, 7121757279798, 65383520552131, 620517308328812, 6079168380979213, 61402851498255790, 638674759049919079, 6833589979500278700
Offset: 0
-
a[n_] := a[n] = 1 + Sum[Binomial[n + 2, k + 3] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 23}]
nmax = 23; A[] = 0; Do[A[x] = 1/(1 - x) + x A[x/(1 - x)]/(1 - x)^4 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
A352862
a(n) = 1 + Sum_{k=0..n-1} binomial(n+3,k+4) * a(k).
Original entry on oeis.org
1, 2, 8, 36, 170, 865, 4742, 27757, 172375, 1130865, 7809057, 56572404, 428710587, 3389749264, 27901667938, 238599540142, 2115876327408, 19425465343555, 184355895494512, 1806122902809371, 18242807108024625, 189750478368293523, 2030261803964224359, 22323607721661782198
Offset: 0
-
a[n_] := a[n] = 1 + Sum[Binomial[n + 3, k + 4] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 23}]
nmax = 23; A[] = 0; Do[A[x] = 1/(1 - x) + x A[x/(1 - x)]/(1 - x)^5 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
A352863
a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n+3,k+3) * a(k).
Original entry on oeis.org
1, 4, 30, 260, 2625, 30296, 393372, 5675160, 90062775, 1559197420, 29242803018, 590638256572, 12781663255725, 295040675093360, 7236113219901240, 187911083837928048, 5150869386839932995, 148622674413214927140, 4502761102131604279590, 142914444471765753144820
Offset: 0
-
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n + 3, k + 3] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 19}]
nmax = 19; CoefficientList[Series[D[x^3/(3! (2 - Exp[x])), {x, 3}], {x, 0, nmax}], x] Range[0, nmax]!
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