{"id":"n-5b44a84154f4b80a","revision":0,"first_index":0,"fingerprint":"f6-9dd229496f4f95ab","name":"shift_x_plus_one of basis_map_A007318 of Triangular array:  row n gives the coefficients of the polynomial p(n,x) defined in Comments.","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"],["9","4"],["49","40","8"],["281","332","128","16"],["1601","2480","1416","352","32"],["9129","17492","13216","4912","896","64"],["52049","118840","111688","55232","15136","2176","128"],["296761","786524","883968","545616","199552","43200","5120","256"],["1692001","5105600","6677320","4940320","2260000","654080","116864","11776","512"],["9647049","32651620","48706720","42006640","23070080","8362304","1999360","303872","26624","1024"],["55003249","206358600","345757320","340562560","218284320","95088000","28497024","5798912","766464","59392","2048"],["313604441","1291700076","2401633920","2659767440","1948840320","991598016","357383168","91202816","16144384","1887232","131072","4096"],["1788035201","8021150160","16387316232","20155869920","16617998880","9672591360","4074275968","1250970112","277791232","43495424","4556800","286720","8192"],["10194593769","49475830388","110168817312","148996319664","136516630400","89463027264","43129291264","15482980096","4137879552","812897280","114073600","10825728","622592","16384"],["58125109649","303429010520","731387312584","1078717345088","1087359526688","792274760320","430214797440","176825915392","55268463104","13069809664","2301380608","292536320","25370624","1343488","32768"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","13","97","757","5881","45709","355249","2760997","21458473","166775293","1296177889","10073896981","78294346201","608503805293","4729292714257"],"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"],["9","4"]],"recurrence":{"denominator":{"coeffs":[["-2"],["2"]]},"terms":[{"coeff":{"coeffs":[["-2"],["-2"],["4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["84","66","12"],["-41","-28","-4"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["82","97","36","4"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":179809,"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-f78ddb7faaf704c4"]},{"id":179810,"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-f78ddb7faaf704c4"]},{"id":179811,"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-f78ddb7faaf704c4"]}],"region_exact":{},"incoming_edges":[{"id":"e-f78ddb7faaf704c4","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-5b44a84154f4b80a","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":"59694c982ff5843726ffe26fa89522232cc6417c803e349402885b8b66656efe","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-20086f90792ae22f","role":"sequence","name":"basis_map_A007318 of Triangular array:  row n gives the coefficients of the polynomial p(n,x) defined in Comments.","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-fd238312a45f35b0","kind":"candidate_equal","node_a_id":"n-5b44a84154f4b80a","node_b_id":"n-b20c37322c58da9b","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-b20c37322c58da9b","other":{"id":"n-b20c37322c58da9b","name":"basis_map_A007318 of basis_map_A007318 of Triangular array:  row n gives the coefficients of the polynomial p(n,x) defined in Comments.","first_index":0,"fingerprint":"f6-9dd229496f4f95ab","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["9","4"],["49","40","8"]],"card_rows":[["1"],["9","4"],["49","40","8"],["281","332","128","16"]],"row_sums_preview":["1","13","97","757","5881","45709","355249","2760997"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e646c30d067149e7","steps":["basis A007318","basis A007318"],"edges":["e-4395b7c136181822","e-b719f34a9b4ca22f"],"text":"A249252 \u2192 basis A007318 \u2192 basis A007318"},"headline":"A249252 \u2192 basis A007318 \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:       9,        4;\n  2:      49,       40,        8;\n  3:     281,      332,      128,       16;\n  4:    1601,     2480,     1416,      352,       32;\n  5:    9129,    17492,    13216,     4912,      896,      64;\n  6:   52049,   118840,   111688,    55232,    15136,    2176,     128;\n  7:  296761,   786524,   883968,   545616,   199552,   43200,    5120,    256;\n  8: 1692001,  5105600,  6677320,  4940320,  2260000,  654080,  116864,  11776,   512;\n  9: 9647049, 32651620, 48706720, 42006640, 23070080, 8362304, 1999360, 303872, 26624, 1024;\n","recurrence_sentence":"The row polynomials satisfy (2*n-2)*P(n,x) = (4*n^2-2*n-2)*P(n-1,x) - (41*n-84+(28*n-66)*x+(4*n-12)*x^2)*d/dx P(n-1,x) + (82+97*x+36*x^2+4*x^3)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 9 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 9 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 2*n - 2)*P[n - 1, x] - (41*n - 84 + (28*n - 66)*x + (4*n - 12)*x^2)*D[P[n - 1, x], x] + (82 + 97*x + 36*x^2 + 4*x^3)*D[P[n - 1, x], {x, 2}])/(2*n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 9 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n^2 - 2*n - 2)*v[i - 1] - (41*n - 84 + (28*n - 66)*x + (4*n - 12)*x^2)*deriv(v[i - 1], x) + (82 + 97*x + 36*x^2 + 4*x^3)*deriv(deriv(v[i - 1], x), x))/(2*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}}],"derivation":{"root":"n-e646c30d067149e7","steps":["basis A007318","x \u21a6 x+1"],"edges":["e-4395b7c136181822","e-f78ddb7faaf704c4"],"text":"A249252 \u2192 basis A007318 \u2192 x \u21a6 x+1"},"headline":"A249252 \u2192 basis A007318 \u2192 x \u21a6 x+1","chain":null}
