{"id":"n-5c39e26cb9304567","revision":0,"first_index":0,"fingerprint":"f6-c9655e7f89d7b43e","name":"Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+2*x+x^2)/(1-x)^4, read by rows.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1","1"],["1","2","1"],["1","7","7","1"],["1","18","42","18","1"],["1","41","198","198","41","1"],["1","88","799","1584","799","88","1"],["1","183","2925","10331","10331","2925","183","1"],["1","374","10056","58874","103310","58874","10056","374","1"],["1","757","33160","305888","869794","869794","305888","33160","757","1"],["1","1524","106293","1488832","6490186","10437528","6490186","1488832","106293","1524","1"],["1","3059","334119","6911965","44361586","108056470","108056470","44361586","6911965","334119","3059","1"],["1","6130","1036006","30989050","284015615","1003231508","1512790580","1003231508","284015615","30989050","1036006","6130","1"],["1","12273","3181578","135352266","1729968575","8575529583","18615386124","18615386124","8575529583","1729968575","135352266","3181578","12273","1"],["1","24560","9704283","579588000","10138717801","68752863248","207487469115","297846177984","207487469115","68752863248","10138717801","579588000","9704283","24560","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","2","4","16","80","480","3360","26880","241920","2419200","26611200","319334400","4151347200","58118860800","871782912000"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A166345","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","1"],["1","2","1"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["-2","0"],["1","-2"],["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-2","2"],["0","1","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":169575,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":false,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0)","region_exact":null,"edges":["e-0fa7bf47b4a608d8","e-dff7465450f7ae66"]},{"id":169576,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-0fa7bf47b4a608d8","e-dff7465450f7ae66"]},{"id":169577,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-0fa7bf47b4a608d8","e-dff7465450f7ae66"]},{"id":169578,"position":3,"claim":"interlacing","from_index":3,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":3,"kind":"weak","orientation":"previous_interlaces_next"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 3","region_exact":null,"edges":["e-0fa7bf47b4a608d8"]},{"id":169579,"position":4,"claim":"interlacing","from_index":4,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":4,"kind":"strict","orientation":"previous_interlaces_next"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m >= 4","region_exact":null,"edges":["e-dff7465450f7ae66"]}],"region_exact":{},"incoming_edges":[{"id":"e-0fa7bf47b4a608d8","operation":{"operation":"theorem_check","theorem":"liu_wang"},"operation_detail":"liu_wang","conclusion_id":"n-5c39e26cb9304567","premise_count":0,"theorem":"L. 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v[1] = 1; v[2] = 1 + x; v[3] = 1 + 2*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n - 2 + (n^2 - 2*n)*x)*v[i - 1] + ((n - 2)*x - (n - 2)*x^2)*deriv(v[i - 1], x))/(n - 2)); 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":"A166345","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: 1,   1;\n   3: 1,   2,     1;\n   4: 1,   7,     7,      1;\n   5: 1,  18,    42,     18,      1;\n   6: 1,  41,   198,    198,     41,      1;\n   7: 1,  88,   799,   1584,    799,     88,      1;\n   8: 1, 183,  2925,  10331,  10331,   2925,    183,     1;\n   9: 1, 374, 10056,  58874, 103310,  58874,  10056,   374,   1;\n  10: 1, 757, 33160, 305888, 869794, 869794, 305888, 33160, 757, 1;\n","recurrence_sentence":"The row polynomials satisfy (n-3)*P(n,x) = (n-3+(n^2-4*n+3)*x)*P(n-1,x) + ((n-3)*x-(n-3)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 1 + x, P(3,x) = 1 + 2*x + x^2.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + x;\nP[3, x_] := 1 + 2*x + x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((n - 3 + (n^2 - 4*n + 3)*x)*P[n - 1, x] + ((n - 3)*x - (n - 3)*x^2)*D[P[n - 1, x], x])/(n - 3)]];\nTable[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; v[3] = 1 + 2*x + x^2; for(n = 4, N, my(i = n); v[i] = ((n - 3 + (n^2 - 4*n + 3)*x)*v[i - 1] + ((n - 3)*x - (n - 3)*x^2)*deriv(v[i - 1], x))/(n - 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}},{"oeis_id":"A166345","target":"reversed_triangle","first_row":0,"oeis_offset":1,"reversed":true,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,   1;\n   3: 1,   2,     1;\n   4: 1,   7,     7,      1;\n   5: 1,  18,    42,     18,      1;\n   6: 1,  41,   198,    198,     41,      1;\n   7: 1,  88,   799,   1584,    799,     88,      1;\n   8: 1, 183,  2925,  10331,  10331,   2925,    183,     1;\n   9: 1, 374, 10056,  58874, 103310,  58874,  10056,   374,   1;\n  10: 1, 757, 33160, 305888, 869794, 869794, 305888, 33160, 757, 1;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where (n-3)*P(n,x) = (n-3+(n^2-4*n+3)*x)*P(n-1,x) + ((n-3)*x-(n-3)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 1 + x, P(3,x) = 1 + 2*x + x^2.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + x;\nP[3, x_] := 1 + 2*x + x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((n - 3 + (n^2 - 4*n + 3)*x)*P[n - 1, x] + ((n - 3)*x - (n - 3)*x^2)*D[P[n - 1, x], x])/(n - 3)]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; v[3] = 1 + 2*x + x^2; for(n = 4, N, my(i = n); v[i] = ((n - 3 + (n^2 - 4*n + 3)*x)*v[i - 1] + ((n - 3)*x - (n - 3)*x^2)*deriv(v[i - 1], x))/(n - 3)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c = Vecrev(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":true,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-5c39e26cb9304567","steps":[],"edges":[],"text":"A166345"},"headline":"A166345: Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+2*x+x^2)/(1-x)^4, read by rows.","chain":null}