{"id":"n-fb270ce887b0e0f1","revision":0,"first_index":0,"fingerprint":"f6-2ee8a94f56aa5e4f","name":"shift_x_plus_one of basis_map_A013611 of Triangle read by rows, s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","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"],["8","4"],["96","80","16"],["1536","1664","576","64"],["30720","39424","18176","3584","256"],["737280","1069056","593920","158720","20480","1024"],["20643840","32882688","20905984","6819840","1208320","110592","4096"],["660602880","1134821376","800522240","301858816","65945600","8372224","573440","16384"],["23781703680","43495981056","33358086144","14069006336","3581476864","565182464","54132736","2883584","65536"],["951268147200","1834966056960","1508307369984","696192598016","199535099904","36933206016","4426039296","331874304","14155776","262144"],["41855798476800","84543579095040","73705388507136","36665703792640","11564314787840","2423201464320","342478553088","32306626560","1950351360","68157440","1048576"],["2009078326886400","4225514990469120","3876032964722688","2054775336075264","701749924986880","162570929438720","26131776405504","2920632287232","222843371520","11072962560","322961408","4194304"],["104472072998092800","227763092811939840","218455774127456256","122352449334804480","44710097443618816","11260688030760960","2009136090841088","256399984558080","23270384467968","1467167539200","61085843456","1509949440","16777216"],["5850436087893196800","13172621489461002240","13144575722385309696","7725560259258875904","2993175254181871616","809438919497089024","157554373210144768","22394943498616832","2328741468438528","175242920067072","9289477390336","328900542464","6979321856","67108864"],["351026165273591808000","813759033719232921600","841365029300962590720","516111918445073793024","210492756287947800576","60539036186552827904","12691018070597042176","1973914102757588992","229304262100779008","19829541077778432","1258340323688448","56891942109184","1734361481216","31943819264","268435456"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","12","192","3840","92160","2580480","82575360","2972712960","118908518400","5231974809600","251134790860800","13059009124761600","731304510986649600","43878270659198976000","2808209322188734464000"],"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":[["4","4"],["4","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":190846,"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-d1a8ca1c8c240ab2"]},{"id":190847,"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-41656559b918afa7","e-d1a8ca1c8c240ab2"]},{"id":190848,"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-41656559b918afa7","e-d1a8ca1c8c240ab2"]},{"id":190849,"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-41656559b918afa7","e-d1a8ca1c8c240ab2"]},{"id":190850,"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-41656559b918afa7"]}],"region_exact":{},"incoming_edges":[{"id":"e-41656559b918afa7","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-fb270ce887b0e0f1","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":"b89ef215f516c10b29e4bab5d1844c21ca1e0ca41a25140578b4f4549e4b2b15","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-f4e656360cd62235","role":"sequence","name":"basis_map_A013611 of Triangle read by rows, s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","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-d1a8ca1c8c240ab2","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-fb270ce887b0e0f1","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":"82a375f23f291ed4cd4293f40b62e97a6b20e6242f2c19561405b1156c1d56ae","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-f4e656360cd62235","role":"sequence","name":"basis_map_A013611 of Triangle read by rows, s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","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-1289bc35541582d9","kind":"fingerprint_related","node_a_id":"n-d5dc1ed6e6bfd018","node_b_id":"n-fb270ce887b0e0f1","fingerprint":"f6-2ee8a94f56aa5e4f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2ee8a94f56aa5e4f"},"other_id":"n-d5dc1ed6e6bfd018","other":{"id":"n-d5dc1ed6e6bfd018","name":"shift_x_plus_one of basis_map_A013610 of Triangle read by rows, s_3(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","first_index":0,"fingerprint":"f6-2ee8a94f56aa5e4f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","3"],["54","45","9"]],"card_rows":[["1"],["6","3"],["54","45","9"],["648","702","243","27"]],"row_sums_preview":["1","9","108","1620","29160","612360","14696640","396809280"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8229dbdd6a4b5cd9","steps":["basis A013610","basis A007318"],"edges":["e-86f8b58a2aa07621","e-149d23f86ba6c4b4"],"text":"A225470 \u2192 basis A013610 \u2192 basis A007318"},"headline":"A225470 \u2192 basis A013610 \u2192 basis A007318","chain":null}},{"id":"l-a35b1139a50ee868","kind":"fingerprint_related","node_a_id":"n-c4e95af1e3477dcf","node_b_id":"n-fb270ce887b0e0f1","fingerprint":"f6-2ee8a94f56aa5e4f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2ee8a94f56aa5e4f"},"other_id":"n-c4e95af1e3477dcf","other":{"id":"n-c4e95af1e3477dcf","name":"reversal of Unsigned 2-Stirling numbers of the first kind.","first_index":0,"fingerprint":"f6-2ee8a94f56aa5e4f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1","5","6"],["1","9","26","24"],["1","14","71","154","120"]],"card_rows":[["1","5","6"],["1","9","26","24"],["1","14","71","154","120"],["1","20","155","580","1044","720"]],"row_sums_preview":["12","60","360","2520","20160","181440","1814400","19958400"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c237629cf5fdbb07","steps":["reverse"],"edges":["e-fc266a6cef1d3b14"],"text":"A143491 \u2192 reverse"},"headline":"A143491 \u2192 reverse","chain":null}},{"id":"l-c9823c60a21f61cc","kind":"fingerprint_related","node_a_id":"n-c237629cf5fdbb07","node_b_id":"n-fb270ce887b0e0f1","fingerprint":"f6-2ee8a94f56aa5e4f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2ee8a94f56aa5e4f"},"other_id":"n-c237629cf5fdbb07","other":{"id":"n-c237629cf5fdbb07","name":"Unsigned 2-Stirling numbers of the first kind.","first_index":0,"fingerprint":"f6-2ee8a94f56aa5e4f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["6","5","1"],["24","26","9","1"],["120","154","71","14","1"]],"card_rows":[["6","5","1"],["24","26","9","1"],["120","154","71","14","1"],["720","1044","580","155","20","1"]],"row_sums_preview":["12","60","360","2520","20160","181440","1814400","19958400"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c237629cf5fdbb07","steps":[],"edges":[],"text":"A143491"},"headline":"Unsigned 2-Stirling numbers of the first kind.","chain":null}},{"id":"l-ccaba174c16c60ff","kind":"fingerprint_related","node_a_id":"n-2feb6dc01f424b0a","node_b_id":"n-fb270ce887b0e0f1","fingerprint":"f6-2ee8a94f56aa5e4f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2ee8a94f56aa5e4f"},"other_id":"n-2feb6dc01f424b0a","other":{"id":"n-2feb6dc01f424b0a","name":"Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....","first_index":0,"fingerprint":"f6-2ee8a94f56aa5e4f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","2"],["1","5","6"]],"card_rows":[["1"],["1","2"],["1","5","6"],["1","9","26","24"]],"row_sums_preview":["1","3","12","60","360","2520","20160","181440"],"regions":["(-inf, 0)","(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":["A145324"],"derivation":{"root":"n-2feb6dc01f424b0a","steps":[],"edges":[],"text":"A145324"},"headline":"A145324: Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....","chain":null}},{"id":"l-e7b4153357c02ea1","kind":"fingerprint_related","node_a_id":"n-914700b1060feeb2","node_b_id":"n-fb270ce887b0e0f1","fingerprint":"f6-2ee8a94f56aa5e4f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2ee8a94f56aa5e4f"},"other_id":"n-914700b1060feeb2","other":{"id":"n-914700b1060feeb2","name":"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)).","first_index":0,"fingerprint":"f6-2ee8a94f56aa5e4f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","1"],["6","5","1"]],"card_rows":[["1"],["2","1"],["6","5","1"],["24","26","9","1"]],"row_sums_preview":["1","3","12","60","360","2520","20160","181440"],"regions":["(-inf, 0)","(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":["A136124"],"derivation":{"root":"n-914700b1060feeb2","steps":[],"edges":[],"text":"A136124"},"headline":"A136124: 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)).","chain":null}},{"id":"l-febb04234c9bccb6","kind":"fingerprint_related","node_a_id":"n-5e3978466b5f73cb","node_b_id":"n-fb270ce887b0e0f1","fingerprint":"f6-2ee8a94f56aa5e4f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2ee8a94f56aa5e4f"},"other_id":"n-5e3978466b5f73cb","other":{"id":"n-5e3978466b5f73cb","name":"shift_x_plus_one of basis_map_A013609 of Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.","first_index":0,"fingerprint":"f6-2ee8a94f56aa5e4f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","2"],["24","20","4"]],"card_rows":[["1"],["4","2"],["24","20","4"],["192","208","72","8"]],"row_sums_preview":["1","6","48","480","5760","80640","1290240","23224320"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4f6e747b794bcc2e","steps":["basis A013609","x \u21a6 x+1"],"edges":["e-03658933e548767a","e-069c80bca774515e"],"text":"A028338 \u2192 basis A013609 \u2192 x \u21a6 x+1"},"headline":"A028338 \u2192 basis A013609 \u2192 x \u21a6 x+1","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:            8,             4;\n  2:           96,            80,            16;\n  3:         1536,          1664,           576,           64;\n  4:        30720,         39424,         18176,         3584,          256;\n  5:       737280,       1069056,        593920,       158720,        20480,        1024;\n  6:     20643840,      32882688,      20905984,      6819840,      1208320,      110592,       4096;\n  7:    660602880,    1134821376,     800522240,    301858816,     65945600,     8372224,     573440,     16384;\n  8:  23781703680,   43495981056,   33358086144,  14069006336,   3581476864,   565182464,   54132736,   2883584,    65536;\n  9: 951268147200, 1834966056960, 1508307369984, 696192598016, 199535099904, 36933206016, 4426039296, 331874304, 14155776, 262144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4*n+4+4*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[(4*n + 4 + 4*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] = (4*n + 4 + 4*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-9c5c208df8605708","steps":["basis A013611","basis A007318"],"edges":["e-8a4919ac2dd1c88f","e-41656559b918afa7"],"text":"A225471 \u2192 basis A013611 \u2192 basis A007318"},"headline":"A225471 \u2192 basis A013611 \u2192 basis A007318","chain":null}