{"id":"n-2bd60c45ab7c2c34","revision":0,"first_index":0,"fingerprint":"f6-31b02e78b9a6fd3c","name":"shift_x_plus_one of basis_map_A013611 of Pascal-(1,5,1) array.","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":[["1"],["6","4"],["61","68","16"],["516","872","448","64"],["4621","10216","7936","2496","256"],["40626","111900","117120","57280","12800","1024"],["359281","1181724","1553040","1033280","362240","62464","4096"],["3171336","12137488","19211136","16186240","7772160","2105344","294912","16384"],["28010041","122238992","226277248","230854784","141099520","52527104","11542528","1359872","65536"],["247343646","1212338036","2569647616","3079116416","2288045056","1087637504","328843264","60637184","6160384","262144"],["2184312901","11879578420","28368948880","39050203520","34189319680","19813173248","7662714880","1944043520","308346880","27525120","1048576"],["19289468556","115270045944","306219958080","476147879360","480120186880","328588156928","155202813952","50407915520","10993008640","1528299520","121634816","4194304"],["170344633861","1109453868408","3245215222656","5625401174080","6421049701120","5071125413888","2833400848384","1125114134528","314179256320","59996897280","7419723776","532676608","16777216"],["1504304517066","10605642265612","33870006680448","64761364080704","82553865162240","73928058948608","47736740233216","22448542367744","7668515602432","1874766069760","318112792576","35412508672","2315255808","67108864"],["13284442948921","100794311049356","348962067079696","729587545011136","1027402913305856","1028982943710208","754390206681088","410434085470208","166162027184128","49706166386688","10793172074496","1646637613056","166614532096","9999220736","268435456"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","10","145","1900","25525","340750","4556125","60895000","813975625","10880031250","145429215625","1943893562500","25983250328125","347307713593750","4642323400703125"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0"],["2"]]},"terms":[{"coeff":{"coeffs":[["42","16"],["-42","-8"],["12","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["188","190","32"],["-68","-64","-8"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["85","148","74","8"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":184759,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-c8bf5d97ba5c321d"]},{"id":184760,"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-c8bf5d97ba5c321d"]},{"id":184761,"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-c8bf5d97ba5c321d"]}],"region_exact":{},"incoming_edges":[{"id":"e-c8bf5d97ba5c321d","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-2bd60c45ab7c2c34","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"32ddd12d5bd3b27789a1ea380ad1c04403ed1ece767924cd3640c16106f90f96","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-590ad09c6151a6a8","role":"sequence","name":"basis_map_A013611 of Pascal-(1,5,1) array.","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":[{"id":"l-b7b11dc25b5a64b6","kind":"candidate_equal","node_a_id":"n-2bd60c45ab7c2c34","node_b_id":"n-4a83e4b905d55ad8","fingerprint":null,"oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"reason":"no all-index equality certificate was found","rows_compared":15},"other_id":"n-4a83e4b905d55ad8","other":{"id":"n-4a83e4b905d55ad8","name":"basis_map_A007318 of basis_map_A013611 of Pascal-(1,5,1) array.","first_index":0,"fingerprint":"f6-31b02e78b9a6fd3c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["6","4"],["61","68","16"]],"card_rows":[["1"],["6","4"],["61","68","16"],["516","872","448","64"]],"row_sums_preview":["1","10","145","1900","25525","340750","4556125","60895000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6ad01dc6e705cdb6","steps":["basis A013611","basis A007318"],"edges":["e-09b3cf169dcfe004","e-296aad118fb9dae2"],"text":"A081580 \u2192 basis A013611 \u2192 basis A007318"},"headline":"A081580 \u2192 basis A013611 \u2192 basis A007318","chain":null}}],"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:         1;\n  1:         6,          4;\n  2:        61,         68,         16;\n  3:       516,        872,        448,         64;\n  4:      4621,      10216,       7936,       2496,        256;\n  5:     40626,     111900,     117120,      57280,      12800,       1024;\n  6:    359281,    1181724,    1553040,    1033280,     362240,      62464,      4096;\n  7:   3171336,   12137488,   19211136,   16186240,    7772160,    2105344,    294912,    16384;\n  8:  28010041,  122238992,  226277248,  230854784,  141099520,   52527104,  11542528,  1359872,   65536;\n  9: 247343646, 1212338036, 2569647616, 3079116416, 2288045056, 1087637504, 328843264, 60637184, 6160384, 262144;\n","recurrence_sentence":"The row polynomials satisfy 2*n*P(n,x) = (12*n^2-42*n+42-(8*n-16)*x)*P(n-1,x) - (68*n-188+(64*n-190)*x+(8*n-32)*x^2)*d/dx P(n-1,x) + (85+148*x+74*x^2+8*x^3)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((12*n^2 - 42*n + 42 - (8*n - 16)*x)*P[n - 1, x] - (68*n - 188 + (64*n - 190)*x + (8*n - 32)*x^2)*D[P[n - 1, x], x] + (85 + 148*x + 74*x^2 + 8*x^3)*D[P[n - 1, x], {x, 2}])/(2*n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((12*n^2 - 42*n + 42 - (8*n - 16)*x)*v[i - 1] - (68*n - 188 + (64*n - 190)*x + (8*n - 32)*x^2)*deriv(v[i - 1], x) + (85 + 148*x + 74*x^2 + 8*x^3)*deriv(deriv(v[i - 1], x), x))/(2*n)); 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-6ad01dc6e705cdb6","steps":["basis A013611","x \u21a6 x+1"],"edges":["e-09b3cf169dcfe004","e-c8bf5d97ba5c321d"],"text":"A081580 \u2192 basis A013611 \u2192 x \u21a6 x+1"},"headline":"A081580 \u2192 basis A013611 \u2192 x \u21a6 x+1","chain":null}