{"id":"n-acacc82d3699f634","revision":0,"first_index":0,"fingerprint":"f6-e0f7dc6dd641b7f6","name":"basis_map_A013611 of Triangle read by rows: The convolution triangle of 2^n, where the convolution triangle of a sequence is defined in A357368.","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":true,"rows":[["1"],["2","8"],["8","48","64"],["32","256","640","512"],["128","1280","4608","7168","4096"],["512","6144","28672","65536","73728","32768"],["2048","28672","163840","491520","819200","720896","262144"],["8192","131072","884736","3276800","7208960","9437184","6815744","2097152"],["32768","589824","4587520","20185088","55050240","95420416","102760448","62914560","16777216"],["131072","2621440","23068672","117440512","381681664","822083584","1174405120","1073741824","570425344","134217728"],["524288","11534336","113246208","654311424","2466250752","6341787648","11274289152","13690208256","10871635968","5100273664","1073741824"],["2097152","50331648","545259520","3523215360","15099494400","45097156608","95831457792","144955146240","153008209920","107374182400","45097156608","8589934592"],["8388608","218103808","2583691264","18454937600","88583700480","301184581632","744103084032","1346472247296","1771674009600","1653562408960","1039382085632","395136991232","68719476736"],["33554432","939524096","12079595520","94489280512","501974302720","1913407930368","5385888989184","11338713661440","17858474016768","20787641712640","17386027614208","9895604649984","3435973836800","549755813888"],["134217728","4026531840","55834574848","474593886208","2763811454976","11669426143232","36850819399680","88441966559232","162143605358592","226018358984704","235845244157952","178670639513600","92908732547072","29686813949952","4398046511104"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","10","120","1440","17280","207360","2488320","29859840","358318080","4299816960","51597803520","619173642240","7430083706880","89161004482560","1069932053790720"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","8"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["-4","-8"],["4","8"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":204877,"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-c477fbeecbbc62a9"]},{"id":204878,"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-c477fbeecbbc62a9"]},{"id":204879,"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-c477fbeecbbc62a9"]},{"id":204880,"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-c477fbeecbbc62a9"]}],"region_exact":{},"incoming_edges":[{"id":"e-c477fbeecbbc62a9","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-acacc82d3699f634","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":"7e413fb08a2c40d7c1c2489d6ea1368536a82f4b6f125cf85bceb72193897dcb","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-ff74e25052457f18","role":"sequence","name":"Triangle read by rows: The convolution triangle of 2^n, where the convolution triangle of a sequence is defined in A357368.","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-1ec3c4ae4157f6c6","kind":"fingerprint_related","node_a_id":"n-f5f6d7397eb76ee7","node_b_id":"n-acacc82d3699f634","fingerprint":"f6-e0f7dc6dd641b7f6","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e0f7dc6dd641b7f6"},"other_id":"n-f5f6d7397eb76ee7","other":{"id":"n-f5f6d7397eb76ee7","name":"basis_map_A013611 of Number of compositions of n with k parts, T(n, k) = binomial(n-1, k-1) for n, k >= 1 and T(n, 0) = 0^n, triangle read by rows for n >= 0 and 0 <= k <= n.","first_index":0,"fingerprint":"f6-e0f7dc6dd641b7f6","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","4"],["2","12","16"]],"card_rows":[["1"],["1","4"],["2","12","16"],["4","32","80","64"]],"row_sums_preview":["1","5","30","180","1080","6480","38880","233280"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3783f6df357ea675","steps":["basis A013611"],"edges":["e-ce077fbc42f0c51e"],"text":"A097805 \u2192 basis A013611"},"headline":"A097805 \u2192 basis A013611","chain":null}},{"id":"l-4baa0d30628b49c3","kind":"fingerprint_related","node_a_id":"n-118ddec8f1e698aa","node_b_id":"n-acacc82d3699f634","fingerprint":"f6-e0f7dc6dd641b7f6","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e0f7dc6dd641b7f6"},"other_id":"n-118ddec8f1e698aa","other":{"id":"n-118ddec8f1e698aa","name":"basis_map_A013611 of Triangle read by rows, T(n,k) = (n+1)*binomial(n-1,k-1).","first_index":0,"fingerprint":"f6-e0f7dc6dd641b7f6","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","8"],["6","36","48"]],"card_rows":[["1"],["2","8"],["6","36","48"],["16","128","320","256"]],"row_sums_preview":["1","10","90","720","5400","38880","272160","1866240"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a88a7983f557eac4","steps":["basis A013611"],"edges":["e-584997da1613bc2e"],"text":"A127952 \u2192 basis A013611"},"headline":"A127952 \u2192 basis A013611","chain":null}},{"id":"l-96a738aed4cb4f28","kind":"fingerprint_related","node_a_id":"n-5b4d475ced54f59f","node_b_id":"n-acacc82d3699f634","fingerprint":"f6-e0f7dc6dd641b7f6","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e0f7dc6dd641b7f6"},"other_id":"n-5b4d475ced54f59f","other":{"id":"n-5b4d475ced54f59f","name":"basis_map_A007318 of (3,1) Pascal triangle.","first_index":0,"fingerprint":"f6-e0f7dc6dd641b7f6","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","1"],["8","6","1"]],"card_rows":[["1"],["4","1"],["8","6","1"],["16","20","8","1"]],"row_sums_preview":["1","5","15","45","135","405","1215","3645"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-92128fbba2409faa","steps":["basis A007318"],"edges":["e-bc413433eedeab55"],"text":"A093560 \u2192 basis A007318"},"headline":"A093560 \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:      2,       8;\n  2:      8,      48,       64;\n  3:     32,     256,      640,       512;\n  4:    128,    1280,     4608,      7168,      4096;\n  5:    512,    6144,    28672,     65536,     73728,     32768;\n  6:   2048,   28672,   163840,    491520,    819200,    720896,     262144;\n  7:   8192,  131072,   884736,   3276800,   7208960,   9437184,    6815744,    2097152;\n  8:  32768,  589824,  4587520,  20185088,  55050240,  95420416,  102760448,   62914560,  16777216;\n  9: 131072, 2621440, 23068672, 117440512, 381681664, 822083584, 1174405120, 1073741824, 570425344, 134217728;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (4*n-4+(8*n-8)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 8*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 8*x;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n - 4 + (8*n - 8)*x)*P[n - 1, x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n - 4 + (8*n - 8)*x)*v[i - 1])/(n - 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-ff74e25052457f18","steps":["basis A013611"],"edges":["e-c477fbeecbbc62a9"],"text":"A380114 \u2192 basis A013611"},"headline":"A380114 \u2192 basis A013611","chain":null}
