{"id":"n-3141285c8fc77c60","revision":0,"first_index":0,"fingerprint":"f6-dd8e883ba3f13bd7","name":"reversal of basis_map_A013610 of Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T(n,k) is also the unsigned Stirling number |s(n+1, k+1)|, denoting the number of permutations on n+1 elements that contain exactly k+1 cycles.","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"],["3","2"],["9","15","6"],["27","81","78","24"],["81","378","639","462","120"],["243","1620","4185","5220","3132","720"],["729","6561","23895","44955","45936","24084","5040"],["2187","25515","124173","326025","497448","439740","207792","40320"],["6561","96228","602154","2095632","4426569","5796252","4581036","1991088","362880"],["19683","354294","2768742","12308436","34236027","61654446","71705628","51783624","20999520","3628800"],["59049","1279395","12203460","67381470","238100877","561559635","893315790","944112780","632618424","241881120","39916800"],["177147","4546773","51963120","348585930","1522880271","4541889429","9418662990","13552127820","13227208632","8317064448","3022323840","479001600"],["531441","15943230","214997409","1721278350","9100257903","33423111810","87300551547","163099002330","215859287556","196904905560","117188809344","40727214720","6227020800"],["1594323","55269864","868197447","8173798776","51398670609","227672946072","729825219981","1711504728648","2930963895288","3612744742464","3108235105872","1762824974976","588862068480","87178291200"],["4782969","189724437","3433640301","37544358033","276802993467","1453998897351","5604569851023","16081892485659","34465462615584","54802692656712","63515876454576","51912001513008","28208960830080","9094465900800","1307674368000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","30","210","1680","15120","151200","1663200","19958400","259459200","3632428800","54486432000","871782912000","14820309504000","266765571072000"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["3","1"],["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":183682,"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-42c6c53bca30515e"]},{"id":183683,"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-42c6c53bca30515e"]},{"id":183684,"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-42c6c53bca30515e"]}],"region_exact":{},"incoming_edges":[{"id":"e-42c6c53bca30515e","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-3141285c8fc77c60","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":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"1f31d1c3d7f924b9eef7e723fae2045ea5ab69cb2423fd1fe04a5dd2b19130cf","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-c89ee0d57229b178","role":"sequence","name":"basis_map_A013610 of Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T(n,k) is also the unsigned Stirling number |s(n+1, k+1)|, denoting the number of permutations on n+1 elements that contain exactly k+1 cycles.","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-9bbb11cc3be7e0ec","kind":"fingerprint_related","node_a_id":"n-3141285c8fc77c60","node_b_id":"n-f6ce73d04c355dca","fingerprint":"f6-dd8e883ba3f13bd7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-dd8e883ba3f13bd7"},"other_id":"n-f6ce73d04c355dca","other":{"id":"n-f6ce73d04c355dca","name":"basis_map_A013610 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-dd8e883ba3f13bd7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","9"],["54","135","81"]],"card_rows":[["1"],["6","9"],["54","135","81"],["648","2106","2187","729"]],"row_sums_preview":["1","15","270","5670","136080","3674160","110224800","3637418400"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8229dbdd6a4b5cd9","steps":["basis A013610","basis A013610"],"edges":["e-86f8b58a2aa07621","e-02a40a7a2efe22d0"],"text":"A225470 \u2192 basis A013610 \u2192 basis A013610"},"headline":"A225470 \u2192 basis A013610 \u2192 basis A013610","chain":null}},{"id":"l-9d1b7437751a4932","kind":"fingerprint_related","node_a_id":"n-3141285c8fc77c60","node_b_id":"n-a6f7ad8574b034ac","fingerprint":"f6-dd8e883ba3f13bd7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-dd8e883ba3f13bd7"},"other_id":"n-a6f7ad8574b034ac","other":{"id":"n-a6f7ad8574b034ac","name":"basis_map_A013609 of basis_map_A013610 of Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of even length, k=0..n.","first_index":0,"fingerprint":"f6-dd8e883ba3f13bd7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","6"],["72","180","108"]],"card_rows":[["1"],["4","6"],["72","180","108"],["2880","9360","9720","3240"]],"row_sums_preview":["1","10","360","25200","2822400","457228800","100590336000","28768836096000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1336d058dca9979e","steps":["basis A013610","basis A013609"],"edges":["e-05481f5802c9cdfa","e-9c3505dcd599244b"],"text":"A204420 \u2192 basis A013610 \u2192 basis A013609"},"headline":"A204420 \u2192 basis A013610 \u2192 basis A013609","chain":null}},{"id":"l-ae6ec953ae310b3f","kind":"fingerprint_related","node_a_id":"n-3141285c8fc77c60","node_b_id":"n-a878b3f1264e21ba","fingerprint":"f6-dd8e883ba3f13bd7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-dd8e883ba3f13bd7"},"other_id":"n-a878b3f1264e21ba","other":{"id":"n-a878b3f1264e21ba","name":"basis_map_A013610 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.","first_index":0,"fingerprint":"f6-dd8e883ba3f13bd7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","12"],["96","240","144"]],"card_rows":[["1"],["8","12"],["96","240","144"],["1536","4992","5184","1728"]],"row_sums_preview":["1","20","480","13440","430080","15482880","619315200","27249868800"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9c5c208df8605708","steps":["basis A013611","basis A013610"],"edges":["e-8a4919ac2dd1c88f","e-82b0c00a778d243a"],"text":"A225471 \u2192 basis A013611 \u2192 basis A013610"},"headline":"A225471 \u2192 basis A013611 \u2192 basis A013610","chain":null}},{"id":"l-bdc101af247b6835","kind":"fingerprint_related","node_a_id":"n-c89ee0d57229b178","node_b_id":"n-3141285c8fc77c60","fingerprint":"f6-dd8e883ba3f13bd7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-dd8e883ba3f13bd7"},"other_id":"n-c89ee0d57229b178","other":{"id":"n-c89ee0d57229b178","name":"basis_map_A013610 of Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T(n,k) is also the unsigned Stirling number |s(n+1, k+1)|, denoting the number of permutations on n+1 elements that contain exactly k+1 cycles.","first_index":0,"fingerprint":"f6-dd8e883ba3f13bd7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","3"],["6","15","9"]],"card_rows":[["1"],["2","3"],["6","15","9"],["24","78","81","27"]],"row_sums_preview":["1","5","30","210","1680","15120","151200","1663200"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-efe23a76953cf69d","steps":["basis A013610"],"edges":["e-4deb1a58186d6d0a"],"text":"A130534 \u2192 basis A013610"},"headline":"A130534 \u2192 basis A013610","chain":null}},{"id":"l-e19ef62361b993a9","kind":"fingerprint_related","node_a_id":"n-3141285c8fc77c60","node_b_id":"n-d5fa840455d1e647","fingerprint":"f6-dd8e883ba3f13bd7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-dd8e883ba3f13bd7"},"other_id":"n-d5fa840455d1e647","other":{"id":"n-d5fa840455d1e647","name":"basis_map_A013610 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-dd8e883ba3f13bd7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","6"],["24","60","36"]],"card_rows":[["1"],["4","6"],["24","60","36"],["192","624","648","216"]],"row_sums_preview":["1","10","120","1680","26880","483840","9676800","212889600"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4f6e747b794bcc2e","steps":["basis A013609","basis A013610"],"edges":["e-03658933e548767a","e-4caeccb95115fd31"],"text":"A028338 \u2192 basis A013609 \u2192 basis A013610"},"headline":"A028338 \u2192 basis A013609 \u2192 basis A013610","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:     3,      2;\n  2:     9,     15,       6;\n  3:    27,     81,      78,       24;\n  4:    81,    378,     639,      462,      120;\n  5:   243,   1620,    4185,     5220,     3132,      720;\n  6:   729,   6561,   23895,    44955,    45936,    24084,     5040;\n  7:  2187,  25515,  124173,   326025,   497448,   439740,   207792,    40320;\n  8:  6561,  96228,  602154,  2095632,  4426569,  5796252,  4581036,  1991088,   362880;\n  9: 19683, 354294, 2768742, 12308436, 34236027, 61654446, 71705628, 51783624, 20999520, 3628800;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+(n+1)*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[(3 + (n + 1)*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] = (3 + (n + 1)*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-efe23a76953cf69d","steps":["basis A013610","reverse"],"edges":["e-4deb1a58186d6d0a","e-42c6c53bca30515e"],"text":"A130534 \u2192 basis A013610 \u2192 reverse"},"headline":"A130534 \u2192 basis A013610 \u2192 reverse","chain":null}