A213500
Rectangular array T(n,k): (row n) = b**c, where b(h) = h, c(h) = h + n - 1, n >= 1, h >= 1, and ** = convolution.
Original entry on oeis.org
1, 4, 2, 10, 7, 3, 20, 16, 10, 4, 35, 30, 22, 13, 5, 56, 50, 40, 28, 16, 6, 84, 77, 65, 50, 34, 19, 7, 120, 112, 98, 80, 60, 40, 22, 8, 165, 156, 140, 119, 95, 70, 46, 25, 9, 220, 210, 192, 168, 140, 110, 80, 52, 28, 10, 286, 275, 255, 228, 196, 161, 125, 90
Offset: 1
Northwest corner (the array is read by southwest falling antidiagonals):
1, 4, 10, 20, 35, 56, 84, ...
2, 7, 16, 30, 50, 77, 112, ...
3, 10, 22, 40, 65, 98, 140, ...
4, 13, 28, 50, 80, 119, 168, ...
5, 16, 34, 60, 95, 140, 196, ...
6, 19, 40, 70, 110, 161, 224, ...
T(6,1) = (1)**(6) = 6;
T(6,2) = (1,2)**(6,7) = 1*7+2*6 = 19;
T(6,3) = (1,2,3)**(6,7,8) = 1*8+2*7+3*6 = 40.
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b[n_] := n; c[n_] := n
t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
r[n_] := Table[t[n, k], {k, 1, 60}] (* A213500 *)
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t(n,k) = sum(i=0, k - 1, (k - i) * (n + i));
tabl(nn) = {for(n=1, nn, for(k=1, n, print1(t(k,n - k + 1),", ");); print(););};
tabl(12) \\ Indranil Ghosh, Mar 26 2017
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def t(n, k): return sum((k - i) * (n + i) for i in range(k))
for n in range(1, 13):
print([t(k, n - k + 1) for k in range(1, n + 1)]) # Indranil Ghosh, Mar 26 2017
A156927
FP3 polynomials related to the generating functions of the columns of the A156921 matrix.
Original entry on oeis.org
1, 1, 1, -6, 29, 31, -283, 245, 298, -286, -108, 119, -3106, 29469, -104585, -220481, 3601363, -15487305, 34949165, -39821950, 4356011, 46881744, -51274736, 9005908, 14663472, -5205168, -1456704, -20736
Offset: 0
The first few rows of the "triangle" of the FP3(z,m) coefficients are:
[1]
[1, 1]
[-6, 29, 31, -283, 245, 298, -286, -108]
The first few FP3 polynomials are:
FP3(z; m=0) = 1
FP3(z; m=1) = (1+z)
FP3(z; m=2) = (-6+29*z+31*z^2-283*z^3+245*z^4+298*z^5-286*z^6-108*z^7)
Some GF3(z;m) are:
GF3(z;m=1) = z^2*(1+z)/((1-z)^4*(1-2*z))
GF3(z;m=2) = z^2*(-6+29*z+31*z^2-283*z^3+245*z^4+298*z^5-286*z^6-108*z^7)/((1-z)^7*(1-2*z)^4*(1-3*z))
For the polynomials in the denominators of the GF3(z;m) see
A157704.
A156929
G.f. of the z^2 coefficients of the FP1 in the third column of the A156921 matrix.
Original entry on oeis.org
-6, -79, -515, -2255, -7321, -17280, -20052, 73530, 683280, 3482667, 14574887, 55023247, 194872885, 661036370, 2174736558, 6996176016, 22133771190, 69145507605, 213941135265, 657095902685
Offset: 2
A156930
G.f. of the z^3 coefficients of the FP1 in the fourth column of the A156921 matrix.
Original entry on oeis.org
119, 1654, 5784, -57421, -974120, -8191112, -51264392, -266367722, -1204269647, -4832097594, -17187443308, -52433219783, -120342975558, -58288009528, 1603731045044, 13940518848356
Offset: 3
A156931
G.f. of the z^4 coefficients of the FP1 in the fifth column of the A156921 matrix.
Original entry on oeis.org
126, 8689, 300930, 5663483, 69028169, 613038531, 4234224501, 23275739871, 98332765273, 250304681662, -554375755759, -13379311589392, -119762221369238, -826135093245122, -4949174987335110
Offset: 3
A213582
Rectangular array: (row n) = b**c, where b(h) = -1 + 2^h, c(h) = n-1+h, n>=1, h>=1, and ** = convolution.
Original entry on oeis.org
1, 5, 2, 16, 9, 3, 42, 27, 13, 4, 99, 68, 38, 17, 5, 219, 156, 94, 49, 21, 6, 466, 339, 213, 120, 60, 25, 7, 968, 713, 459, 270, 146, 71, 29, 8, 1981, 1470, 960, 579, 327, 172, 82, 33, 9, 4017, 2994, 1972, 1207, 699, 384, 198, 93, 37, 10, 8100, 6053, 4007, 2474, 1454, 819, 441, 224, 104, 41, 11
Offset: 1
Northwest corner (the array is read by falling antidiagonals):
1...5....16...42....99....219
2...9....27...68....156...339
3...13...38...94....213...459
4...17...49...120...270...579
5...21...60...146...327...699
6...25...71...172...384...819
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Flat(List([1..12], n-> List([1..n], k-> 2*(k+1)*(2^(n-k+1) -1) -(n-k+1)*(n+k+4)/2 ))); # G. C. Greubel, Jul 08 2019
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[[2*(k+1)*(2^(n-k+1) -1) -(n-k+1)*(n+k+4)/2: k in [1..n]]: n in [1..12]]; // G. C. Greubel, Jul 08 2019
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(* First program *)
b[n_]:= 2^n - 1; c[n_]:= n;
T[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
TableForm[Table[T[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[T[n-k+1, k], {n, 12}, {k, n, 1, -1}]] (* A213582 *)
r[n_]:= Table[T[n, k], {k, 40}]
Table[T[n, n], {n, 1, 40}] (* A213583 *)
s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A156928 *)
(* Second program *)
Table[2*(k+1)*(2^(n-k+1) -1) -(n-k+1)*(n+k+4)/2, {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Jul 08 2019 *)
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t(n,k) = 2*(k+1)*(2^(n-k+1) -1) -(n-k+1)*(n+k+4)/2;
for(n=1,12, for(k=1,n, print1(t(n,k), ", "))) \\ G. C. Greubel, Jul 08 2019
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[[2*(k+1)*(2^(n-k+1) -1) -(n-k+1)*(n+k+4)/2 for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jul 08 2019
A213583
Principal diagonal of the convolution array A213582.
Original entry on oeis.org
1, 9, 38, 120, 327, 819, 1948, 4482, 10085, 22341, 48930, 106236, 229075, 491175, 1048184, 2227782, 4718097, 9960921, 20970910, 44039520, 92273951, 192937179, 402652308, 838859850, 1744829437, 3623877549, 7516191578, 15569255172, 32212253355, 66571991631
Offset: 1
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List([1..40], n-> (n+1)*(2^(n+2) -3*n-4)/2); # G. C. Greubel, Jul 08 2019
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[(n+1)*(2^(n+2) -3*n-4)/2: n in [1..40]]; // G. C. Greubel, Jul 08 2019
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(* First program *)
b[n_]:= 2^n - 1; c[n_]:= n;
T[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
TableForm[Table[T[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[T[n-k+1, k], {n, 12}, {k, n, 1, -1}]] (* A213582 *)
r[n_]:= Table[T[n, k], {k, 40}] (* columns of antidiagonal triangle *)
Table[T[n, n], {n, 1, 40}] (* A213583 *)
s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A156928 *)
(* Second program *)
LinearRecurrence[{7,-19,25,-16,4},{1,9,38,120,327},40] (* Harvey P. Dale, Apr 06 2013 *)
Table[(n+1)*(2^(n+2)-3*n-4)/2, {n,40}] (* G. C. Greubel, Jul 08 2019 *)
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Vec(x*(1 + 2*x - 6*x^2) / ((1 - x)^3*(1 - 2*x)^2) + O(x^40)) \\ Colin Barker, Nov 04 2017
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vector(40, n, (n+1)*(2^(n+2) -3*n-4)/2) \\ G. C. Greubel, Jul 08 2019
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[(n+1)*(2^(n+2) -3*n-4)/2 for n in (1..40)] # G. C. Greubel, Jul 08 2019
Showing 1-7 of 7 results.
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