{"id":"n-5fc6172728a1754c","revision":0,"first_index":0,"fingerprint":"f6-2fb91d8b80545e88","name":"basis_map_A013609 of Triangular T(n,k) = T(n-1, k) + T(n-1, k-1) + 5*T(n-2, k-1), read by rows.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["2"],["10","10"],["50","100","8"],["250","750","516","16"],["750","3000","3572","1144","32"],["2750","13750","23224","14672","2512","64"],["9250","55500","121808","117232","45968","5472","128"],["32250","225750","608236","783680","485680","128320","11840","256"],["110750","886000","2832012","4588072","3940880","1715040","335680","25472","512"],["382750","3444750","12734704","24840928","27323104","16810240","5443840","841984","54528","1024"],["1319250","13192500","55378968","126411744","169913184","136250688","63336960","16055808","2050304","116224","2048"],["4552250","50074750","235263956","615133104","977674656","969609984","594496896","217433856","44904704","4883456","246784","4096"],["15700750","188409000","979497252","2886642520","5299298880","6274954560","4807405440","2337510144","695486720","120660480","11432960","522240","8192"],["54162750","704115750","4012879784","13160584624","27411587120","37773303360","34833304320","21321969408","8464855808","2105758720","314255360","26398720","1101824","16384"],["186829250","2615609500","16215567328","58575113936","136507263088","214737542560","231999788160","172072152576","86426185472","28699398656","6103797760","798248960","60264448","2318336","32768"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["2","20","158","1532","8498","56972","355358","2276012","14434418","91877852","584027678","3714278492","23617529138","150184293932","955000112798"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["2"],["10","10"],["50","100","8"],["250","750","516","16"]],"recurrence":{"denominator":{"coeffs":[["-3"],["1"]]},"terms":[{"coeff":{"coeffs":[["-15","-30"],["5","10"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-6","-6"],["2","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":199532,"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":true,"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-856dc4b7af9f5bee"]},{"id":199533,"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-856dc4b7af9f5bee"]},{"id":199534,"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-856dc4b7af9f5bee"]},{"id":199535,"position":3,"claim":"interlacing","from_index":3,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":3,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 3","region_exact":null,"edges":["e-856dc4b7af9f5bee"]}],"region_exact":{},"incoming_edges":[{"id":"e-856dc4b7af9f5bee","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-5fc6172728a1754c","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"ba21337cc277ce0427853760049063d1212abb9ecdf38b9c2d2bfb9d50df56f6","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":3,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-a44eea8c1f4cf4e5","role":"sequence","name":"Triangular T(n,k) = T(n-1, k) + T(n-1, k-1) + 5*T(n-2, k-1), read by rows.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:      2;\n  1:     10,      10;\n  2:     50,     100,        8;\n  3:    250,     750,      516,       16;\n  4:    750,    3000,     3572,     1144,       32;\n  5:   2750,   13750,    23224,    14672,     2512,       64;\n  6:   9250,   55500,   121808,   117232,    45968,     5472,     128;\n  7:  32250,  225750,   608236,   783680,   485680,   128320,   11840,    256;\n  8: 110750,  886000,  2832012,  4588072,  3940880,  1715040,  335680,  25472,   512;\n  9: 382750, 3444750, 12734704, 24840928, 27323104, 16810240, 5443840, 841984, 54528, 1024;\n","recurrence_sentence":"The row polynomials satisfy (n-3)*P(n,x) = (5*n-15+(10*n-30)*x)*P(n-2,x) + (2*n-6+(2*n-6)*x)*P(n-1,x) for n >= 4, with P(0,x) = 2, P(1,x) = 10 + 10*x, P(2,x) = 50 + 100*x + 8*x^2, P(3,x) = 250 + 750*x + 516*x^2 + 16*x^3.","mathematica":"P[0, x_] := 2;\nP[1, x_] := 10 + 10*x;\nP[2, x_] := 50 + 100*x + 8*x^2;\nP[3, x_] := 250 + 750*x + 516*x^2 + 16*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[((5*n - 15 + (10*n - 30)*x)*P[n - 2, x] + (2*n - 6 + (2*n - 6)*x)*P[n - 1, x])/(n - 3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 10 + 10*x; v[3] = 50 + 100*x + 8*x^2; v[4] = 250 + 750*x + 516*x^2 + 16*x^3; for(n = 4, N, my(i = n + 1); v[i] = ((5*n - 15 + (10*n - 30)*x)*v[i - 2] + (2*n - 6 + (2*n - 6)*x)*v[i - 1])/(n - 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}}],"derivation":{"root":"n-a44eea8c1f4cf4e5","steps":["basis A013609"],"edges":["e-856dc4b7af9f5bee"],"text":"A153518 \u2192 basis A013609"},"headline":"A153518 \u2192 basis A013609","chain":null}