A051340 -id:A051340 - cp's OEIS frontend

A185874 Second accumulation array of A051340, by antidiagonals.

1, 3, 4, 6, 11, 10, 10, 21, 26, 20, 15, 34, 48, 50, 35, 21, 50, 76, 90, 85, 56, 28, 69, 110, 140, 150, 133, 84, 36, 91, 150, 200, 230, 231, 196, 120, 45, 116, 196, 270, 325, 350, 336, 276, 165, 55, 144, 248, 350, 435, 490, 504, 468, 375, 220, 66, 175, 306, 440, 560, 651, 700, 696, 630, 495, 286, 78, 209, 370, 540, 700, 833, 924, 960, 930, 825, 638, 364, 91, 246, 440, 650, 855, 1036, 1176, 1260, 1275, 1210, 1056, 806, 455, 105, 286, 516, 770, 1025, 1260, 1456, 1596, 1665, 1650, 1540, 1326, 1001, 560

Offset: 1

Clark Kimberling, Feb 05 2011

A member of the accumulation chain: A051340 < A141419 < A185874 < A185875 < A185876 < ... (See A144112 for the definition of accumulation array.)

			Northwest corner:
.   1,   3,   6,   10,   15,   21,   28,   36,   45,   55, ...
.   4,  11,  21,   34,   50,   69,   91,  116,  144,  175, ...
.  10,  26,  48,   76,  110,  150,  196,  248,  306,  370, ...
.  20,  50,  90,  140,  200,  270,  350,  440,  540,  650, ...
.  35,  85, 150,  230,  325,  435,  560,  700,  855, 1025, ...
.  56, 133, 231,  350,  490,  651,  833, 1036, 1260, 1505, ...
.  84, 196, 336,  504,  700,  924, 1176, 1456, 1764, 2100, ...
. 120, 276, 468,  696,  960, 1260, 1596, 1968, 2376, 2820, ...
. 165, 375, 630,  930, 1275, 1665, 2100, 2580, 3105, 3675, ...
. 220, 495, 825, 1210, 1650, 2145, 2695, 3300, 3960, 4675, ...
...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Johann Cigler, Some elementary observations on Narayana polynomials and related topics, arXiv:1611.05252 [math.CO], 2016. See p. 24.

Cf. A144112, A051340, A141419, A185875, A185876.
Row 1 to 5: A000217, A115056, 2*A140096, 10*A000096, 5*A059845.
Column 1 to 3: A000292, A051925, A267370 and 3*A005581.
Main diagonal: A117066.

f[n_, k_] := (1/12)*k*n*(1 + n)*(1 + 3*k + 2*n);
TableForm[Table[f[n, k], {n, 1, 10}, {k, 1, 15}]]
Table[f[n - k + 1, k], {n, 14}, {k, n, 1, -1}] // Flatten

T(n,k) = k*n*(n+1)*(2*n+3*k+1)/12 for k>=1, n>=1.

Edited by Bruno Berselli, Jan 14 2016

A185875 Third accumulation array of A051340, by antidiagonals.

Original entry on oeis.org

1, 4, 5, 10, 19, 15, 20, 46, 55, 35, 35, 90, 130, 125, 70, 56, 155, 250, 290, 245, 126, 84, 245, 425, 550, 560, 434, 210, 120, 364, 665, 925, 1050, 980, 714, 330, 165, 516, 980, 1435, 1750, 1820, 1596, 1110, 495, 220, 705, 1380, 2100, 2695, 3010, 2940, 2460, 1650, 715, 286, 935, 1875, 2940, 3920, 4606, 4830, 4500, 3630, 2365, 1001, 364, 1210, 2475, 3975, 5460, 6664, 7350, 7350, 6600, 5170

Offset: 1

Clark Kimberling, Feb 05 2011

A member of the accumulation chain A051340 < A141419 < A185874 < A185875 < A185876 < ... (See A144112 for the definition of accumulation array.)

			Northwest corner:
   1,   4,  10,  20,  35
   5,  19,  46,  90, 155
  15,  55, 130, 250, 425
  35, 125, 290, 550, 925

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Johann Cigler, Some elementary observations on Narayana polynomials and related topics, arXiv:1611.05252 [math.CO], 2016. See p. 24.

Cf. A051340, A141419, A144112, A185875, A185876.

Row 1: A000292; Column 1: A000332.

f[n_,k_]:= k*(1+k)*n*(1+n)*(2+n)*(5+4*k+3*n)/144;
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]]
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten

T(n,k) = (3*n+4*k+5)*C(k,2)*C(n,3)/12, k>=1, n>=1.

A185876 Fourth accumulation array of A051340, by antidiagonals.

Original entry on oeis.org

1, 5, 6, 15, 29, 21, 35, 85, 99, 56, 70, 195, 285, 259, 126, 126, 385, 645, 735, 574, 252, 210, 686, 1260, 1645, 1610, 1134, 462, 330, 1134, 2226, 3185, 3570, 3150, 2058, 792, 495, 1770, 3654, 5586, 6860, 6930, 5670, 3498, 1287, 715, 2640, 5670, 9114, 11956, 13230, 12390, 9570, 5643, 2002, 1001, 3795, 8415, 14070

Offset: 1

Clark Kimberling, Feb 05 2011

A member of the accumulation chain A051340 < A141419 < A185874 < A185875 < A185876 < ... (See A144112 for the definition of accumulation array.)

			Northwest corner:
   1,   5,  15,   35,   70
   6,  29,  85,  195,  385
  21,  99, 285,  645, 1260
  56, 259, 735, 1645, 3185

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Johann Cigler, Some elementary observations on Narayana polynomials and related topics, arXiv:1611.05252 [math.CO], 2016. See p. 24.

Cf. A051340, A141419, A144112, A185874, A185875.

Row 1: A000332, column 1: A000389.

f[n_,k_]:=k(1+k)n(1+n)(2+n)(5+4k+3n)/144;
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]] (* A185875 *)
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}]; (* accumulation array of {f(n,k)} *)
Factor[s[n,k]]  (* formula for A185876 *)
TableForm[Table[s[n,k],{n,1,10},{k,1,15}]] (* A185876 *)
Table[s[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten

T(n,k) = (4*n+5*k+11)*C(k+2,3)*C(n+4,4)/20, k>=1, n>=1.

A131914 3A002024 - 2A051340.

Original entry on oeis.org

1, 4, 2, 7, 5, 3, 10, 8, 6, 4, 13, 11, 9, 7, 5, 16, 14, 12, 10, 8, 6, 19, 17, 15, 13, 11, 9, 7, 22, 20, 18, 16, 14, 12, 10, 8, 25, 23, 21, 19, 17, 15, 13, 11, 9, 28, 26, 24, 22, 20, 18, 16, 14, 12, 10

Offset: 1

Gary W. Adamson, Jul 27 2007

Row sums = the hexagonal numbers, A000384: (1, 6, 15, 28, 45, ...).
From Boris Putievskiy, Jan 24 2013: (Start)
Table T(n,k) = n + 3*k - 3, n, k > 0, read by antidiagonals. General case A209304. Let m be a positive integer. The first column of the table T(n,1) is the sequence of the positive integers A000027. Every subsequent column is formed from the previous column, shifted by m elements.
For m=0 the result is A002260,
for m=1 the result is A002024,
for m=2 the result is A128076,
for m=3 the result is A131914,
for m=4 the result is A209304. (End)

			First few rows of the triangle:
   1;
   4,  2;
   7,  5,  3;
  10,  8,  6,  4;
  13, 11,  9,  7,  5;
  16, 14, 12, 10,  8,  6;
  19, 17, 15, 13, 11,  9,  7;
  ...

Boris Putievskiy, Transformations [Of] Integer Sequences And Pairing Functions, arXiv preprint arXiv:1212.2732 [math.CO], 2012.

Cf. A002024, A051340, A000384, A003056, A002260, A002024, A128076, A209304.

3*A002024 - 2*A051340 as infinite lower triangular matrices.
From Boris Putievskiy, Jan 24 2013: (Start)
For the general case
a(n) = m*A003056 - (m-1)*A002260.
a(n) = m*(t+1) + (m-1)*(t*(t+1)/2-n), where t = floor((-1+sqrt(8*n-7))/2).
For m = 3,
a(n) = 3*A003056 - 2*A002260.
a(n) = 3*(t+1) + 2*(t*(t+1)/2-n), where t = floor((-1+sqrt(8*n-7))/2). (End)

A127949 A000012 as an infinite lower triangular matrix with all 1's; A127899 = a simple transform; then A000012 * A127899. Given A051340, change all 1's to -1. Triangle read by rows, (n-1) -1's followed by "n".

Original entry on oeis.org

1, -1, 2, -1, -1, 3, -1, -1, -1, 4, -1, -1, -1, -1, 5, -1, -1, -1, -1, -1, 6, -1, -1, -1, -1, -1, -1, 7, -1, -1, -1, -1, -1, -1, -1, 8, -1, -1, -1, -1, -1, -1, -1, -1, 9, -1, -1, -1, -1, -1, -1, -1, -1, -1, 10, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 11, -1, -1, -1, -1, -1, -1

Offset: 1

Gary W. Adamson, Feb 09 2007

For the inverse of A127949 see A126615, a harmonic triangle.

This is one way to define an inverse to A000217. - R. J. Mathar, Apr 30 2010

			First few rows of the triangle are:
1;
-1, 2;
-1, -1, 3;
-1, -1, -1, 4;
...

Cf. A000012, A127899, A051340, A126615.

A127949 := proc(n) if issqr(1+8*n) then (sqrt(1+8*n)-1)/2 ; else -1 ; end if; end proc: seq(A127949(n),n=1..120) ; # R. J. Mathar, Apr 30 2010

More terms from R. J. Mathar, Apr 30 2010

A130298 A051340 * A130296.

Original entry on oeis.org

1, 5, 2, 12, 4, 3, 22, 6, 5, 4, 35, 8, 7, 6, 5, 51, 10, 9, 8, 7, 6, 70, 12, 11, 10, 9, 8, 7, 92, 14, 13, 12, 11, 10, 9, 8, 117, 16, 15, 14, 13, 12, 11, 10, 9, 145, 18, 17, 16, 15, 14, 13, 12, 11, 10

Offset: 1

Gary W. Adamson, May 20 2007

Row sums = A003215: (1, 7, 19, 37, 61, 91, ...).
Left border = A000326: (1, 5, 12, 22, 35, ...).
A130299 = A130296 * A051340.

			First few rows of the triangle:
   1;
   5,  2;
  12,  4,  3;
  22,  6,  5,  4;
  35,  8,  7,  6, 5;
  51, 10,  9,  8, 7, 6;
  70, 12, 11, 10, 9, 8, 7;
  ...

Cf. A051340, A130296, A003215, A000326.

A051340 * A130296 as infinite lower triangular matrices.

A131034 A129686 * A051340.

Original entry on oeis.org

1, 2, 1, 4, 1, 1, 6, 2, 1, 1, 8, 2, 2, 1, 1, 10, 2, 2, 2, 1, 12, 2, 2, 2, 2, 1, 1, 14, 2, 2, 2, 2, 2, 1, 1, 16, 2, 2, 2, 2, 2, 2, 1, 1

Offset: 0

Gary W. Adamson, Jun 10 2007

Row sums = A113127: (1, 3, 6, 10, 14, 18, 22, ...).

			First few rows of the triangle:
   1;
   2, 1;
   4, 1, 1;
   6, 2, 1, 1;
   8, 2, 2, 1, 1;
  10, 2, 2, 2, 1, 1;
  ...

Cf. A129686, A051340, A113127.

A129686 * A051340 as infinite lower triangular matrices.

A131228 3A051340 - 2A128174.

Original entry on oeis.org

1, 3, 4, 1, 3, 7, 3, 1, 3, 10, 1, 3, 1, 3, 13, 3, 1, 3, 1, 3, 16, 1, 3, 1, 3, 1, 3, 19, 3, 1, 3, 1, 3, 1, 3, 22, 1, 3, 1, 3, 1, 3, 25, 3, 1, 3, 1, 3, 1, 3, 1, 3, 28

Offset: 0

Gary W. Adamson, Jun 20 2007

nonn

Row sums = A131229.

			First few rows of the triangle are:
1;
3, 4;
1, 3, 7;
3, 1, 3, 10;
1, 3, 1, 3, 13;
...

Cf. A051340, A128174, A131229, A131227.

3*A051340 - 2*A128174 as infinite lower triangular matrices.

A130265 Triangle read by rows: matrix product A007318 * A051340.

Original entry on oeis.org

1, 2, 2, 4, 5, 3, 8, 10, 10, 4, 16, 19, 23, 17, 5, 32, 36, 46, 46, 26, 6, 64, 69, 87, 102, 82, 37, 7, 128, 134, 162, 204, 204, 134, 50, 8, 256, 263, 303, 387, 443, 373, 205, 65, 9, 512, 520, 574, 718, 886, 886, 634, 298, 82, 10

Offset: 0

Gary W. Adamson, May 18 2007

			First few rows of the triangle are:
   1;
   2,  2;
   4,  5,  3;
   8, 10, 10,   4;
  16, 19, 23,  17,  5;
  32, 36, 46,  46, 26,  6;
  64, 69, 87, 102, 82, 37,  7;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A001787 (row sums), A007318, A051340, A058877, A084633, A258431.

A130265:= func< n,k | k eq n select n+1 else (k+1)*Binomial(n,k) + (&+[Binomial(n, j+k): j in [1..n-k]]) >;
[A130265(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 18 2023

A051340 := proc(n,k)
    if k = n then
        n+1 ;
    elif k <= n then
        1;
    else
        0;
    end if;
end proc:
A130265 := proc(n,k)
    add( binomial(n,j)*A051340(j,k),j=k..n) ;
end proc:
seq(seq(A130265(n,k),k=0..n),n=0..15) ; # R. J. Mathar, Aug 06 2016

T[n_, k_]:= (k+1)*Binomial[n,k] + Binomial[n,k+1]*Hypergeometric2F1[1, k-n+1, k+2, -1];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 18 2023 *)

def A130265(n,k): return (k+1)*binomial(n,k) + sum(binomial(n, j+k) for j in range(1,n-k+1))
flatten([[A130265(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Mar 18 2023

Binomial transform of A051340.
From G. C. Greubel, Mar 18 2023: (Start)
T(n, k) = (k+1)*binomial(n,k) + Sum_{j=1..n-k} binomial(n, j+k).
T(n, k) = (k+1)*binomial(n,k) + binomial(n,k+1)*Hypergeometric2F1([1, k-n+1], [k+2], -1).
T(2*n, n) = (1/2)*T(2*n+1, n) = A258431(n+1).
Sum_{k=0..n} T(n, k) = A001787(n+1).
Sum_{k=0..n-1} T(n, k) = A058877(n+1), for n >= 1.
Sum_{k=0..n} (-1)^k*T(n, k) = (-1)^n*A084633(n). (End)

Missing term inserted by R. J. Mathar, Aug 06 2016

A130266 A051340 * A128174.

Original entry on oeis.org

1, 1, 2, 4, 1, 3, 2, 5, 1, 4, 7, 2, 6, 1, 5, 3, 8, 2, 7, 1, 6, 10, 3, 9, 2, 8, 1, 7, 4, 11, 3, 10, 2, 9, 1, 8, 13, 4, 12, 3, 11, 2, 10, 1, 9, 5, 14, 4, 13, 3, 12, 2, 11, 1, 10, 16, 5, 15, 4, 14, 3, 13, 2, 12, 1, 11, 6, 17, 5, 16, 4, 15, 3, 14, 2, 13

Offset: 0

Gary W. Adamson, May 18 2007

Row sums = A014255: (1, 3, 8, 12, 21, 27, 40, ...).

Left border = A123684: (1, 1, 4, 2, 7, 3, 10, 4, ...).

			First few rows of the triangle:
   1;
   1, 2;
   4, 1, 3;
   2, 5, 1, 4;
   7, 2, 6, 1, 5;
   3, 8, 2, 7, 1, 6;
  10, 3, 9, 2, 8, 1, 7;
  ...

Cf. A051340, A128174, A014255, A123684.

A128174 := proc(n,k)
    if k > n or k < 1 then
        0;
    else
        modp(k+n+1,2) ;
    end if;
end proc:
A051340 := proc(n,k)
    if k = n then
        n ;
    elif k <= n then
        1;
    else
        0;
    end if;
end proc:
A130266 := proc(n,k)
    add( A051340(n,j)*A128174(j,k),j=k..n) ;
end proc:
seq(seq(A130266(n,k),k=1..n),n=1..15) ; # R. J. Mathar, Aug 06 2016

A051340 * A128174 as infinite lower triangular matrices.

cp's OEIS Frontend

A185874 Second accumulation array of A051340, by antidiagonals.

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A185875 Third accumulation array of A051340, by antidiagonals.

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A185876 Fourth accumulation array of A051340, by antidiagonals.

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A131914 3*A002024 - 2*A051340.

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A127949 A000012 as an infinite lower triangular matrix with all 1's; A127899 = a simple transform; then A000012 * A127899. Given A051340, change all 1's to -1. Triangle read by rows, (n-1) -1's followed by "n".

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A130298 A051340 * A130296.

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A131034 A129686 * A051340.

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A131228 3*A051340 - 2*A128174.

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A130265 Triangle read by rows: matrix product A007318 * A051340.

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A131914 3A002024 - 2A051340.

A131228 3A051340 - 2A128174.