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
A001296
4-dimensional pyramidal numbers: a(n) = (3*n+1)*binomial(n+2, 3)/4. Also Stirling2(n+2, n).
Original entry on oeis.org
0, 1, 7, 25, 65, 140, 266, 462, 750, 1155, 1705, 2431, 3367, 4550, 6020, 7820, 9996, 12597, 15675, 19285, 23485, 28336, 33902, 40250, 47450, 55575, 64701, 74907, 86275, 98890, 112840, 128216, 145112, 163625, 183855, 205905, 229881, 255892, 284050, 314470
Offset: 0
G.f. = x + 7*x^2 + 25*x^3 + 65*x^4 + 140*x^5 + 266*x^6 + 462*x^7 + 750*x^8 + 1155*x^9 + ...
- M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 835.
- A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 195.
- L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 227, #16.
- S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 166, Table 10.4/I/3).
- F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 223.
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- T. D. Noe, Table of n, a(n) for n = 0..1000
- M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
- P. Aluffi, Degrees of projections of rank loci, arXiv:1408.1702 [math.AG], 2014. ["After compiling the results of many explicit computations, we noticed that many of the numbers d_{n,r,S} appear in the existing literature in contexts far removed from the enumerative geometry of rank conditions; we owe this surprising (to us) observation to perusal of [Slo14]."]
- S. Butler and P. Karasik, A note on nested sums, J. Int. Seq. 13 (2010), 10.4.4, page 5.
- M. Griffiths, Remodified Bessel Functions via Coincidences and Near Coincidences, Journal of Integer Sequences, Vol. 14 (2011), Article 11.7.1.
- L. Hogben, Choice and Chance by Cardpack and Chessboard, Vol. 1, Chanticleer Press, NY, 1950, p. 36.
- C. Krishnamachaki, The operator (xD)^n, J. Indian Math. Soc., 15 (1923),3-4. [Annotated scanned copy]
- Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 29.
- T. Mansour, Restricted permutations by patterns of type 2-1, arXiv:math/0202219 [math.CO], 2002.
- Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
- Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
- Eric Weisstein's World of Mathematics, Stirling numbers of the 2nd kind.
- Index to sequences related to pyramidal numbers
- Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
a(n)=f(n, 2) where f is given in
A034261.
a(n)=
A093560(n+3, 4), (3, 1)-Pascal column.
Cf.
A220212 for a list of sequences produced by the convolution of the natural numbers with the k-gonal numbers.
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/* A000027 convolved with A000326: */ A000326:=func; [&+[(n-i+1)*A000326(i): i in [0..n]]: n in [0..40]]; // Bruno Berselli, Dec 06 2012
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[(3*n+1)*Binomial(n+2,3)/4: n in [0..40]]; // Vincenzo Librandi, Jul 30 2014
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A001296:=-(1+2*z)/(z-1)**5; # Simon Plouffe in his 1992 dissertation for sequence without the leading zero
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Table[n*(1+n)*(2+n)*(1+3*n)/24, {n, 0, 100}]
CoefficientList[Series[x (1 + 2 x)/(1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Jul 30 2014 *)
Table[StirlingS2[n+2, n], {n, 0, 40}] (* Jean-François Alcover, Jun 24 2015 *)
Table[ListCorrelate[Accumulate[Range[n]],Range[n]],{n,0,40}]//Flatten (* or *) LinearRecurrence[{5,-10,10,-5,1},{0,1,7,25,65},40] (* Harvey P. Dale, Aug 14 2017 *)
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t(n)=n*(n+1)/2
for(i=1,30,print1(","sum(j=1,i,j*t(j))))
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{a(n) = n * (n+1) * (n+2) * (3*n+1) / 24}; /* Michael Somos, Sep 04 2017 */
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[stirling_number2(n+2,n) for n in range(0,38)] # Zerinvary Lajos, Mar 14 2009
A213819
Rectangular array: (row n) = b**c, where b(h) = h, c(h) = 3*n-4+3*h, n>=1, h>=1, and ** = convolution.
Original entry on oeis.org
2, 9, 5, 24, 18, 8, 50, 42, 27, 11, 90, 80, 60, 36, 14, 147, 135, 110, 78, 45, 17, 224, 210, 180, 140, 96, 54, 20, 324, 308, 273, 225, 170, 114, 63, 23, 450, 432, 392, 336, 270, 200, 132, 72, 26, 605, 585, 540, 476, 399, 315
Offset: 1
Northwest corner (the array is read by falling antidiagonals):
2....9....24....50....90....147
5....18...42....80....135...210
8....27...60....110...180...273
11...36...78....140...225...336
14...45...96....170...270...399
17...54...114...200...315...462
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b[n_]:=n;c[n_]:=3n-1;
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}] (* A213819 *)
Table[t[n,n],{n,1,40}] (* A213820 *)
d/2 (* A002414 *)
s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
Table[s[n],{n,1,50}] (* A153978 *)
s1/2 (* A001296 *)
A247237
Triangle read by rows: T(n,k) is the coefficient in the transformation Sum_{k=0..n} (k+1)*x^k = Sum_{k=0..n} T(n,k)*(x-k)^k.
Original entry on oeis.org
1, 3, 2, 3, 14, 3, 3, 50, 39, 4, 3, 130, 279, 84, 5, 3, 280, 1479, 984, 155, 6, 3, 532, 6519, 8544, 2675, 258, 7, 3, 924, 25335, 61464, 34035, 6138, 399, 8, 3, 1500, 89847, 388056, 356595, 106938, 12495, 584, 9, 3, 2310, 297207, 2225136, 3259635, 1524438, 284655, 23264, 819, 10
Offset: 0
From _Wolfdieter Lang_, Jan 14 2015: (Start)
The triangle T(n,k) starts:
n\k 0 1 2 3 4 5 6 7 8 9 ...
0: 1
1: 3 2
2: 3 14 3
3: 3 50 39 4
4: 3 130 279 84 5
5: 3 280 1479 984 155 6
6: 3 532 6519 8544 2675 258 7
7: 3 924 25335 61464 34035 6138 399 8
8: 3 1500 89847 388056 356595 106938 12495 584 9
9: 3 2310 297207 2225136 3259635 1524438 284655 23264 819 10
...
-----------------------------------------------------------------
n = 3: 1 + 2*x + 3*x^2 + 4*x^3 = 3*(x-0)^0 + 50*(x-1)^1 + 39*(x-2)^2 + 4*(x-3)^3.
(End)
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T(n,k)=(k+1)-sum(i=k+1,n,(-i)^(i-k)*binomial(i,k)*T(n,i))
for(n=0,10,for(k=0,n,print1(T(n,k),", ")))
Showing 1-4 of 4 results.
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