A382472 -id:A382472 - cp's OEIS frontend

A108479 Antidiagonal sums of number triangle A086645.

1, 1, 2, 7, 17, 44, 117, 305, 798, 2091, 5473, 14328, 37513, 98209, 257114, 673135, 1762289, 4613732, 12078909, 31622993, 82790070, 216747219, 567451585, 1485607536, 3889371025, 10182505537, 26658145586, 69791931223, 182717648081

Offset: 0

Paul Barry, Jun 04 2005

Seiichi Manyama, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (2,1,2,-1).

Cf. A005252, A051286, A086645, A182890.
Cf. A381421, A382230, A382470, A382471, A382472, A382473, A382474.
Cf. A376729, A376730, A376731.

LinearRecurrence[{2,1,2,-1},{1,1,2,7},30] (* Harvey P. Dale, Jun 01 2021 *)

G.f.: (1 - x - x^2)/(1 - 2*x - x^2 - 2*x^3 + x^4).
a(n) = 2*(n-1) + a(n-2) + 2*a(n-3) - a(n-4).
a(n) = Sum_{k=0..floor(n/2)} C(2*(n-k), 2*k).
a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-k} C(2*(n-2*k), j) * C(2*k, j).
a(n) = A005252(2*n). - Seiichi Manyama, Aug 11 2024

A381421 a(n) = Sum_{k=0..n} (k+1) * binomial(2k,2n-2*k).

Original entry on oeis.org

1, 2, 5, 22, 68, 206, 631, 1870, 5467, 15836, 45416, 129260, 365565, 1028122, 2877697, 8021010, 22274476, 61653850, 170152275, 468347046, 1286055927, 3523777912, 9635982160, 26302324504, 71674754873, 195015074610, 529846108989, 1437657038030, 3896050721940

Offset: 0

Seiichi Manyama, Mar 28 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (4,-2,0,-11,0,-2,4,-1).

Cf. A108479, A382230, A382470, A382471, A382472, A382473, A382474.

Cf. A034839.

[&+[(k+1) * Binomial(2*k, 2*n-2*k): k in [0..n]]: n in [0..29]]; // Vincenzo Librandi, Apr 23 2025

Table[Sum[(k+1)*Binomial[2*k,2*n-2*k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, Apr 23 2025 *)

a(n) = sum(k=0, n, (k+1)*binomial(2*k, 2*n-2*k));

my(N=1, M=30, x='x+O('x^M), X=1-x-x^2, Y=3); Vec(sum(k=0, (N+1)\2, 4^k*binomial(N+1, 2*k)*X^(N+1-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1))

G.f.: ((1-x-x^2)^2 + 4*x^3) / ((1-x-x^2)^2 - 4*x^3)^2.

a(n) = 4*a(n-1) - 2*a(n-2) - 11*a(n-4) - 2*a(n-6) + 4*a(n-7) - a(n-8).

A382230 a(n) = Sum_{k=0..n} binomial(k+2,2) * binomial(2k,2n-2*k).

Original entry on oeis.org

1, 3, 9, 46, 171, 591, 2033, 6714, 21606, 68308, 212370, 651234, 1974113, 5924277, 17623671, 52025858, 152539077, 444530073, 1288396257, 3715833732, 10668907932, 30507914696, 86912853588, 246755125332, 698353551105, 1970673504951, 5545952371509, 15568330002486

Offset: 0

Seiichi Manyama, Mar 28 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (6,-9,2,-18,30,7,30,-18,2,-9,6,-1).

Cf. A108479, A381421, A382470, A382471, A382472, A382473, A382474.

Cf. A034839, A377145.

[&+[Binomial(k+2, 2) * Binomial(2*k, 2*n-2*k): k in [0..n]]: n in [0..29]]; // Vincenzo Librandi, Apr 22 2025

Table[Sum[Binomial[k+2,2]*Binomial[2*k,2*n-2*k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, Apr 22 2025 *)

a(n) = sum(k=0, n, binomial(k+2, 2)*binomial(2*k, 2*n-2*k));

my(N=2, M=30, x='x+O('x^M), X=1-x-x^2, Y=3); Vec(sum(k=0, (N+1)\2, 4^k*binomial(N+1, 2*k)*X^(N+1-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1))

G.f.: (Sum_{k=0..1} 4^k * binomial(3,2*k) * (1-x-x^2)^(3-2*k) * x^(3*k)) / ((1-x-x^2)^2 - 4*x^3)^3.

a(n) = 6*a(n-1) - 9*a(n-2) + 2*a(n-3) - 18*a(n-4) + 30*a(n-5) + 7*a(n-6) + 30*a(n-7) - 18*a(n-8) + 2*a(n-9) - 9*a(n-10) + 6*a(n-11) - a(n-12).

A382470 a(n) = Sum_{k=0..n} binomial(k+3,3) * binomial(2k,2n-2*k).

Original entry on oeis.org

1, 4, 14, 80, 345, 1336, 5074, 18404, 64460, 220276, 736242, 2415128, 7798043, 24833160, 78131242, 243211412, 749926963, 2292771088, 6956262660, 20959406680, 62753991192, 186809711448, 553172044548, 1630068765840, 4781871397429, 13969460520764

Offset: 0

Seiichi Manyama, Mar 28 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (8,-20,16,-26,88,-48,24,-163,24,-48,88,-26,16,-20,8,-1).

Cf. A108479, A381421, A382230, A382471, A382472, A382473, A382474.

Cf. A034839, A377148.

[&+[Binomial(k+3,3) * Binomial(2*k,2*n-2*k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Apr 10 2025

Table[Sum[Binomial[k+3,3]*Binomial[2*k,2*n-2*k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, Apr 10 2025 *)

a(n) = sum(k=0, n, binomial(k+3, 3)*binomial(2*k, 2*n-2*k));

my(N=3, M=30, x='x+O('x^M), X=1-x-x^2, Y=3); Vec(sum(k=0, (N+1)\2, 4^k*binomial(N+1, 2*k)*X^(N+1-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1))

G.f.: (Sum_{k=0..2} 4^k * binomial(4,2*k) * (1-x-x^2)^(4-2*k) * x^(3*k)) / ((1-x-x^2)^2 - 4*x^3)^4.

a(n) = 8*a(n-1) - 20*a(n-2) + 16*a(n-3) - 26*a(n-4) + 88*a(n-5) - 48*a(n-6) + 24*a(n-7) - 163*a(n-8) + 24*a(n-9) - 48*a(n-10) + 88*a(n-11) - 26*a(n-12) + 16*a(n-13) - 20*a(n-14) + 8*a(n-15) - a(n-16).

A382471 a(n) = Sum_{k=0..n} binomial(k+4,4) * binomial(2k,2n-2*k).

Original entry on oeis.org

1, 5, 20, 125, 610, 2611, 10815, 42610, 161005, 590155, 2106362, 7348265, 25141430, 84569395, 280246795, 916465742, 2961805180, 9470735650, 29994694130, 94172180660, 293326457342, 907028460410, 2786036875580, 8505001839950, 25815678641935, 77945771624609

Offset: 0

Seiichi Manyama, Mar 28 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (10,-35,50,-55,172,-250,100,-365,510,-29,510,-365,100,-250,172,-55,50,-35,10,-1).

Cf. A108479, A381421, A382230, A382470, A382472, A382473, A382474.

Cf. A034839, A377152.

[&+[Binomial(k+4,4) * Binomial(2*k,2*n-2*k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Apr 10 2025

Table[Sum[Binomial[k+4,4]*Binomial[2*k,2*n-2*k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, Apr 10 2025 *)

a(n) = sum(k=0, n, binomial(k+4, 4)*binomial(2*k, 2*n-2*k));

my(N=4, M=30, x='x+O('x^M), X=1-x-x^2, Y=3); Vec(sum(k=0, (N+1)\2, 4^k*binomial(N+1, 2*k)*X^(N+1-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1))

G.f.: (Sum_{k=0..2} 4^k * binomial(5,2*k) * (1-x-x^2)^(5-2*k) * x^(3*k)) / ((1-x-x^2)^2 - 4*x^3)^5.

A382473 a(n) = Sum_{k=0..n} binomial(k+6,6) * binomial(2k,2n-2*k).

Original entry on oeis.org

1, 7, 35, 252, 1498, 7602, 36498, 165600, 713769, 2957647, 11850223, 46111352, 174956250, 649284286, 2362771938, 8449241836, 29744151416, 103237104740, 353744829032, 1198001464940, 4013905507150, 13316690882670, 43780154987030, 142726581203640

Offset: 0

Seiichi Manyama, Mar 28 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (14, -77, 210, -336, 602, -1435, 2018, -1981, 4312, -5894, 3360, -7721, 9562, -3079, 9562, -7721, 3360, -5894, 4312, -1981, 2018, -1435, 602, -336, 210, -77, 14, -1).

Cf. A108479, A381421, A382230, A382470, A382471, A382472, A382474.

Cf. A034839, A377158.

[&+[Binomial(k+6, 6) * Binomial(2*k, 2*n-2*k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Apr 11 2025

Table[Sum[Binomial[k+6,6]*Binomial[2*k,2*n-2*k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, Apr 11 2025 *)

a(n) = sum(k=0, n, binomial(k+6, 6)*binomial(2*k, 2*n-2*k));

my(N=6, M=30, x='x+O('x^M), X=1-x-x^2, Y=3); Vec(sum(k=0, (N+1)\2, 4^k*binomial(N+1, 2*k)*X^(N+1-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1))

G.f.: (Sum_{k=0..3} 4^k * binomial(7,2*k) * (1-x-x^2)^(7-2*k) * x^(3*k)) / ((1-x-x^2)^2 - 4*x^3)^7.

A382474 a(n) = Sum_{k=0..n} binomial(k+7,7) * binomial(2k,2n-2*k).

Original entry on oeis.org

1, 8, 44, 336, 2166, 11832, 60576, 292248, 1334817, 5840296, 24637976, 100684376, 400255050, 1553016960, 5897388492, 21967711160, 80425346844, 289868771928, 1029979010972, 3612517052608, 12520285820362, 42919328903928, 145643017892472, 489606988741128

Offset: 0

Seiichi Manyama, Mar 28 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (16, -104, 352, -708, 1232, -2800, 5168, -6122, 9728, -18008, 16816, -19152, 37744, -27096, 24960, -50611, 24960, -27096, 37744, -19152, 16816, -18008, 9728, -6122, 5168, -2800, 1232, -708, 352, -104, 16, -1).

Cf. A108479, A381421, A382230, A382470, A382471, A382472, A382473.

Cf. A034839, A377159.

[&+[Binomial(k+7, 7) * Binomial(2*k, 2*n-2*k): k in [0..n]]: n in [0..29]]; // Vincenzo Librandi, Apr 22 2025

Table[Sum[Binomial[k+7,7]*Binomial[2*k,2*n-2*k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, Apr 22 2025 *)

a(n) = sum(k=0, n, binomial(k+7, 7)*binomial(2*k, 2*n-2*k));

my(N=7, M=30, x='x+O('x^M), X=1-x-x^2, Y=3); Vec(sum(k=0, (N+1)\2, 4^k*binomial(N+1, 2*k)*X^(N+1-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1))

G.f.: (Sum_{k=0..4} 4^k * binomial(8,2*k) * (1-x-x^2)^(8-2*k) * x^(3*k)) / ((1-x-x^2)^2 - 4*x^3)^8.

cp's OEIS Frontend

A108479 Antidiagonal sums of number triangle A086645.

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A381421 a(n) = Sum_{k=0..n} (k+1) * binomial(2*k,2*n-2*k).

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A382230 a(n) = Sum_{k=0..n} binomial(k+2,2) * binomial(2*k,2*n-2*k).

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A382470 a(n) = Sum_{k=0..n} binomial(k+3,3) * binomial(2*k,2*n-2*k).

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A382471 a(n) = Sum_{k=0..n} binomial(k+4,4) * binomial(2*k,2*n-2*k).

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A382473 a(n) = Sum_{k=0..n} binomial(k+6,6) * binomial(2*k,2*n-2*k).

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A382474 a(n) = Sum_{k=0..n} binomial(k+7,7) * binomial(2*k,2*n-2*k).

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A381421 a(n) = Sum_{k=0..n} (k+1) * binomial(2k,2n-2*k).

A382230 a(n) = Sum_{k=0..n} binomial(k+2,2) * binomial(2k,2n-2*k).

A382470 a(n) = Sum_{k=0..n} binomial(k+3,3) * binomial(2k,2n-2*k).

A382471 a(n) = Sum_{k=0..n} binomial(k+4,4) * binomial(2k,2n-2*k).

A382473 a(n) = Sum_{k=0..n} binomial(k+6,6) * binomial(2k,2n-2*k).

A382474 a(n) = Sum_{k=0..n} binomial(k+7,7) * binomial(2k,2n-2*k).