{"id":"n-42e9b771fcf14435","revision":0,"first_index":0,"fingerprint":"f6-2ae3ed455d3e70cf","name":"basis_map_A013610 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","12"],["31","96","72"],["145","648","936","432"],["673","3936","8424","7776","2592"],["3127","22596","64008","88560","59616","15552"],["14527","124992","440496","812160","824256","435456","93312"],["67489","673680","2839536","6541344","8880192","7091712","3079296","559872"],["313537","3561792","17471664","48282048","82114560","87899904","57760128","21275136","3359232"],["1456615","18554364","103818096","334619424","684039168","918843264","810228096","451816704","144446976","20155392"],["6767071","95523744","600384312","2211062400","5279909184","8535995136","9454651776","7079021568","3426416640","967458816","120932352"],["31438129","487100184","3397460184","14073322320","38445845184","72684200448","96978321792","91260815616","59324037120","25355483136","6409414656","725594112"],["146053729","2464203168","18886737240","86913543456","267329416608","578699313408","903598416000","1024878168576","837614421504","480827031552","183938107392","42084458496","4353564672"],["678529303","12383064372","103447149624","523579325328","1790576396640","4367501425728","7813629712512","10376755547904","10225263702528","7381005327360","3792075761664","1312599748608","274274574336","26121388032"]],"degrees":[0,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["1","1","19","199","2161","23401","253459","2745199","29733121","322038001","3487978099","37778122999","409173032881","4431733435801","47999891653939"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["3","6"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["4","6"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":202206,"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-27bb478e099b9d3f"]},{"id":202207,"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-27bb478e099b9d3f"]},{"id":202208,"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-27bb478e099b9d3f"]},{"id":202209,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-27bb478e099b9d3f"]}],"region_exact":{},"incoming_edges":[{"id":"e-27bb478e099b9d3f","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-42e9b771fcf14435","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":"31b44f0548f76589c7954c1cf38f77fb06d04bdce79f348ae17897ba382bb0a5","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"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":[],"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,      12;\n  3:     31,      96,       72;\n  4:    145,     648,      936,      432;\n  5:    673,    3936,     8424,     7776,     2592;\n  6:   3127,   22596,    64008,    88560,    59616,    15552;\n  7:  14527,  124992,   440496,   812160,   824256,   435456,    93312;\n  8:  67489,  673680,  2839536,  6541344,  8880192,  7091712,  3079296,   559872;\n  9: 313537, 3561792, 17471664, 48282048, 82114560, 87899904, 57760128, 21275136, 3359232;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+6*x)*P(n-2,x) + (4+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(3 + 6*x)*P[n - 2, x] + (4 + 6*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (3 + 6*x)*v[i - 2] + (4 + 6*x)*v[i - 1]); 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","basis A013610"],"edges":["e-064ccd10ffc87509","e-27bb478e099b9d3f"],"text":"A035607 \u2192 basis A013609 \u2192 basis A013610"},"headline":"A035607 \u2192 basis A013609 \u2192 basis A013610","chain":null}