{"id":"n-9681de120ff6e17e","revision":0,"first_index":0,"fingerprint":"f6-1bf4f6258e36ae6b","name":"shift_x_plus_one of basis_map_A013609 of Square array A(n,r), the number of neighbors at a sharp Manhattan distance r in a finite n-hypercube lattice, read by upwards antidiagonals; A(n,r) = Sum_{k=0..min(n,r)} binomial(r-1,k-1)*binomial(n,k)* 2^k.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"irregular","degree_slope":null,"degree_intercept":null,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["7","4"],["31","32","8"],["145","216","104","16"],["673","1312","936","288","32"],["3127","7532","7112","3280","736","64"],["14527","41664","48944","30080","10176","1792","128"],["67489","224560","315504","242272","109632","29184","4224","256"],["313537","1187264","1941296","1788224","1013760","361728","79232","9728","512"],["1456615","6184788","11535344","12393312","8444928","3781248","1111424","206592","22016","1024"],["6767071","31841248","66709368","81891200","65184064","35127552","12969344","3236864","522240","49152","2048"],["31438129","162366728","377495576","521234160","474640064","299111936","133029248","41728768","9041920","1288192","108544","4096"],["146053729","821401056","2098526360","3219020128","3300363168","2381478656","1239504000","468622848","127665664","24428544","3115008","237568","8192"],["678529303","4127688124","11494127736","19391826864","22105881440","17973256896","10718284928","4744744192","1558491648","374993920","64219136","7409664","516096","16384"]],"degrees":[0,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["1","1","11","71","481","3241","21851","147311","993121","6695281","45137291","304300151","2051487361","13830424921","93239986331"],"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"],["1"],["7","4"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["2"],["-5"],["2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["41","39","8"],["-14","-12","-2"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["21","32","15","2"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":176239,"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-19035e6825c5d565"]},{"id":176240,"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-19035e6825c5d565"]},{"id":176241,"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-19035e6825c5d565"]}],"region_exact":{},"incoming_edges":[{"id":"e-19035e6825c5d565","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-9681de120ff6e17e","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":"a038a74780ee7d6150eac61edb8c1519c7da0445a95af07750a7d74eea86d73a","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-45d316ccbe3d57d1","role":"sequence","name":"basis_map_A013609 of Square array A(n,r), the number of neighbors at a sharp Manhattan distance r in a finite n-hypercube lattice, read by upwards antidiagonals; A(n,r) = Sum_{k=0..min(n,r)} binomial(r-1,k-1)*binomial(n,k)* 2^k.","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-db825097d964877f","kind":"candidate_equal","node_a_id":"n-9681de120ff6e17e","node_b_id":"n-39f1ca0006673cbd","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-39f1ca0006673cbd","other":{"id":"n-39f1ca0006673cbd","name":"basis_map_A007318 of basis_map_A013609 of Square array A(n,r), the number of neighbors at a sharp Manhattan distance r in a finite n-hypercube lattice, read by upwards antidiagonals; A(n,r) = Sum_{k=0..min(n,r)} binomial(r-1,k-1)*binomial(n,k)* 2^k.","first_index":0,"fingerprint":"f6-1bf4f6258e36ae6b","polytool_family":null,"degree_pattern":"irregular","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["7","4"]],"card_rows":[["1"],["1"],["7","4"],["31","32","8"]],"row_sums_preview":["1","1","11","71","481","3241","21851","147311"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e073e5e0b021f36a","steps":["basis A013609","basis A007318"],"edges":["e-064ccd10ffc87509","e-2c7a01782dad3994"],"text":"A035607 \u2192 basis A013609 \u2192 basis A007318"},"headline":"A035607 \u2192 basis A013609 \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:      1;\n  2:      7,       4;\n  3:     31,      32,       8;\n  4:    145,     216,     104,      16;\n  5:    673,    1312,     936,     288,      32;\n  6:   3127,    7532,    7112,    3280,     736,     64;\n  7:  14527,   41664,   48944,   30080,   10176,   1792,   128;\n  8:  67489,  224560,  315504,  242272,  109632,  29184,  4224,  256;\n  9: 313537, 1187264, 1941296, 1788224, 1013760, 361728, 79232, 9728, 512;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = (2*n^2-5*n+2)*P(n-1,x) - (14*n-41+(12*n-39)*x+(2*n-8)*x^2)*d/dx P(n-1,x) + (21+32*x+15*x^2+2*x^3)*d^2/dx^2 P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 7 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[2, x_] := 7 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n^2 - 5*n + 2)*P[n - 1, x] - (14*n - 41 + (12*n - 39)*x + (2*n - 8)*x^2)*D[P[n - 1, x], x] + (21 + 32*x + 15*x^2 + 2*x^3)*D[P[n - 1, x], {x, 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] = 1; v[3] = 7 + 4*x; for(n = 3, N, my(i = n + 1); v[i] = ((2*n^2 - 5*n + 2)*v[i - 1] - (14*n - 41 + (12*n - 39)*x + (2*n - 8)*x^2)*deriv(v[i - 1], x) + (21 + 32*x + 15*x^2 + 2*x^3)*deriv(deriv(v[i - 1], x), 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}}],"derivation":{"root":"n-e073e5e0b021f36a","steps":["basis A013609","x \u21a6 x+1"],"edges":["e-064ccd10ffc87509","e-19035e6825c5d565"],"text":"A035607 \u2192 basis A013609 \u2192 x \u21a6 x+1"},"headline":"A035607 \u2192 basis A013609 \u2192 x \u21a6 x+1","chain":null}