{"id":"n-f860d7ad52328c75","revision":0,"first_index":0,"fingerprint":"f6-e7fd69580c549563","name":"shift_x_plus_one of basis_map_A007318 of Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[x, 0] = 1, p[x, -1] = 0, p[x, 1] = m*x and m = 2. Triangle read by rows, for n >= 0 and 0 <= k <= n.","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"],["4","2"],["20","18","4"],["120","148","60","8"],["840","1276","716","176","16"],["6720","11888","8280","2840","480","32"],["60480","120432","98296","42120","10000","1248","64"],["604800","1325280","1223824","617792","184240","32480","3136","128"],["6652800","15787680","16112624","9243360","3262224","725760","99456","7680","256"],["79833600","202757760","224926848","143145568","57633408","15233568","2644992","291072","18432","512"],["1037836800","2795518080","3329564544","2310746080","1035525440","313303200","64852032","9073920","821760","43520","1024"],["14529715200","41212926720","52204939776","39009574208","19118848320","6457295680","1534534848","256738944","29652480","2252800","101376","2048"],["217945728000","647253331200","865499950080","689553492672","364801873216","135097131840","35932614080","6920153856","958265088","93096960","6026240","233472","4096"],["3487131648000","10791944755200","15142505863680","12763855782912","7215936956800","2891157855872","845116088960","182587689856","29172549120","3406081536","282613760","15788032","532480","8192"],["59281238016000","190437324134400","279006489192960","247270560036864","148198639831424","63581557463424","20149289224064","4794222905472","861108714752","116248484352","11616596992","833624064","40628224","1204224","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","42","336","3024","30240","332640","3991680","51891840","726485760","10897286400","174356582400","2964061900800","53353114214400","1013709170073600"],"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":{"terms":[{"coeff":{"coeffs":[["3","2"],["1","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":181705,"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-14d39f2fc12becc9"]},{"id":181706,"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-14d39f2fc12becc9","e-592f60604f5dbee6","e-6a96c1059c15d050"]},{"id":181707,"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-14d39f2fc12becc9","e-592f60604f5dbee6","e-6a96c1059c15d050"]},{"id":181708,"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-14d39f2fc12becc9","e-592f60604f5dbee6","e-6a96c1059c15d050"]},{"id":181709,"position":4,"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-592f60604f5dbee6","e-6a96c1059c15d050"]}],"region_exact":{},"incoming_edges":[{"id":"e-592f60604f5dbee6","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-f860d7ad52328c75","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":"f4ba7645501c183eb9af09e95cfb6a87574e2a2e160d26781628f26fcf0eaa04","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-7a2a38f140d4a893","role":"sequence","name":"basis_map_A007318 of Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[x, 0] = 1, p[x, -1] = 0, p[x, 1] = m*x and m = 2. Triangle read by rows, for n >= 0 and 0 <= k <= n.","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]"}]},{"id":"e-6a96c1059c15d050","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-f860d7ad52328c75","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":"95ae26dad1532a72b66700652d8412693010b84b4bd1b550f5d56b45c899dde1","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-634bc9ca21d13c4b","role":"sequence","name":"basis_map_A007318 of Triangle read by rows: T(n,k) = (-1)^(n+k)*Sum_{j=1..k} s(n,j), where s(n,j) are the signed Stirling numbers of the first kind (n >= 2; 1 <= k <= n-1; s(n,j) = A008275(n,j)).","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]"}]},{"id":"e-14d39f2fc12becc9","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-f860d7ad52328c75","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":"a84cd595e35bbb1ddf20b81454e8517b2eca0d87e2ec90a9ea3c55511b3684e9","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-7a2a38f140d4a893","role":"sequence","name":"basis_map_A007318 of Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[x, 0] = 1, p[x, -1] = 0, p[x, 1] = m*x and m = 2. Triangle read by rows, for n >= 0 and 0 <= k <= n.","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-60461407ffcd9b04","kind":"fingerprint_related","node_a_id":"n-f860d7ad52328c75","node_b_id":"n-b70ac42dd252f00b","fingerprint":"f6-e7fd69580c549563","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e7fd69580c549563"},"other_id":"n-b70ac42dd252f00b","other":{"id":"n-b70ac42dd252f00b","name":"basis_map_A013609 of basis_map_A007318 of Unsigned 2-Stirling numbers of the first kind.","first_index":0,"fingerprint":"f6-e7fd69580c549563","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["20","18","4"],["120","148","60","8"],["840","1276","716","176","16"]],"card_rows":[["20","18","4"],["120","148","60","8"],["840","1276","716","176","16"],["6720","11888","8280","2840","480","32"]],"row_sums_preview":["42","336","3024","30240","332640","3991680","51891840","726485760"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c237629cf5fdbb07","steps":["basis A007318","basis A013609"],"edges":["e-a33c0ab38a356260","e-31671906239038d4"],"text":"A143491 \u2192 basis A007318 \u2192 basis A013609"},"headline":"A143491 \u2192 basis A007318 \u2192 basis A013609","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:        4,         2;\n  2:       20,        18,         4;\n  3:      120,       148,        60,         8;\n  4:      840,      1276,       716,       176,       16;\n  5:     6720,     11888,      8280,      2840,      480,       32;\n  6:    60480,    120432,     98296,     42120,    10000,     1248,      64;\n  7:   604800,   1325280,   1223824,    617792,   184240,    32480,    3136,    128;\n  8:  6652800,  15787680,  16112624,   9243360,  3262224,   725760,   99456,   7680,   256;\n  9: 79833600, 202757760, 224926848, 143145568, 57633408, 15233568, 2644992, 291072, 18432, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n+3+2*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(n + 3 + 2*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; for(n = 1, N, my(i = n + 1); v[i] = (n + 3 + 2*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-60c7e36cbe887b7e","steps":["x \u21a6 x+1","x \u21a6 x+1"],"edges":["e-01af55cf26557a61","e-14d39f2fc12becc9"],"text":"A137320 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x+1"},"headline":"A137320 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x+1","chain":null}
