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coefficients of one sign stay of one sign","lean_route":null,"lean_declarations":["RealRooted.DegreeDropReversal.splits_X_pow_mul_iff","Polynomial.Splits.iterate_derivative"],"certificate_hash":"2e328668c98fc5fa271037fd38c6fffaad254d8a0fdd48e4d0d75d8d6f083ad4","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-efe23a76953cf69d","role":"sequence","name":"Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. 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v[1] = 1; v[2] = 3 + 3*x; v[3] = 11 + 18*x + 6*x^2; for(n = 3, N, my(i = n + 1); v[i] = (3*n + 3*x)*v[i - 1] - (3*n^2 - 3*n + 1 + (6*n - 3)*x + 3*x^2)*v[i - 2] + (n^3 - 3*n^2 + 3*n - 1 + (3*n^2 - 6*n + 3)*x + (3*n - 3)*x^2 + x^3)*v[i - 3]); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}},{"oeis_id":"A163932","target":"triangle","first_row":0,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:        1;\n   2:        3,        3;\n   3:       11,       18,        6;\n   4:       50,      105,       60,       10;\n   5:      274,      675,      510,      150,       15;\n   6:     1764,     4872,     4410,     1750,      315,      21;\n   7:    13068,    39396,    40614,    19600,     4830,     588,     28;\n   8:   109584,   354372,   403704,   224490,    68040,   11466,   1008,    36;\n   9:  1026576,  3518100,  4342080,  2693250,   949095,  198450,  24360,  1620,   45;\n  10: 10628640, 38260728, 50457000, 34169300, 13530825, 3313233, 508200, 47520, 2475, 55;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3*n-3+3*x)*P(n-1,x) - (3*n^2-9*n+7+(6*n-9)*x+3*x^2)*P(n-2,x) + (n^3-6*n^2+12*n-8+(3*n^2-12*n+12)*x+(3*n-6)*x^2+x^3)*P(n-3,x) for n >= 4, with P(1,x) = 1, P(2,x) = 3 + 3*x, P(3,x) = 11 + 18*x + 6*x^2.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 3 + 3*x;\nP[3, x_] := 11 + 18*x + 6*x^2;\nP[n_, x_] := P[n, x] = Expand[(3*n - 3 + 3*x)*P[n - 1, x] - (3*n^2 - 9*n + 7 + (6*n - 9)*x + 3*x^2)*P[n - 2, x] + (n^3 - 6*n^2 + 12*n - 8 + (3*n^2 - 12*n + 12)*x + (3*n - 6)*x^2 + x^3)*P[n - 3, x]];\nTable[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + 3*x; v[3] = 11 + 18*x + 6*x^2; for(n = 4, N, my(i = n); v[i] = (3*n - 3 + 3*x)*v[i - 1] - (3*n^2 - 9*n + 7 + (6*n - 9)*x + 3*x^2)*v[i - 2] + (n^3 - 6*n^2 + 12*n - 8 + (3*n^2 - 12*n + 12)*x + (3*n - 6)*x^2 + x^3)*v[i - 3]); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":0,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-efe23a76953cf69d","steps":["D\u00b2\u2218x (shift +1)"],"edges":["e-337faa044bd16e1a"],"text":"A130534 \u2192 D\u00b2\u2218x (shift +1)"},"headline":"A163932: Triangle related to the asymptotic expansion of E(x,m=3,n).","chain":"A130534 \u2192 D\u00b2\u2218x (shift +1)"}