{"id":"n-381741d6edcf9397","revision":0,"first_index":0,"fingerprint":"f6-01d09cfb9e6b0947","name":"mobius_x_over_x_plus_one of basis_map_A008277 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":false,"rows":[["1"],["2","3"],["6","18","13"],["24","108","156","73"],["120","720","1560","1460","501"],["720","5400","15600","21900","15030","4051"],["5040","45360","163800","306600","315630","170142","37633"],["40320","423360","1834560","4292400","5891760","4763976","2107448","394353"],["362880","4354560","22014720","61810560","106051680","114335424","75868128","28393416","4596553"],["3628800","48988800","283046400","927158400","1908930240","2572547040","2276043840","1277703720","413689770","58941091"],["39916800","598752000","3891888000","14569632000","34997054400","56596034880","62591205600","46849136400","22752937350","6483520010","824073141"],["479001600","7903526400","57081024000","240398928000","659944454400","1245112767360","1652407827840","1546021501200","1001129243400","427912320660","108777654612","12470162233"],["6227020800","112086374400","890463974400","4166914752000","12868916860800","27748227386880","42962603523840","48235870837440","39044040492600","22251440674320","8484657059736","1945345308348","202976401213"],["87178291200","1699976678400","14733131212800","75837848486400","260238096518400","631272173051520","1117027691619840","1463154748735680","1421203073930640","1012440550681560","514735861623984","177026423059668","36941705020766","3535017524403"],["1307674368000","27461161728000","257829796224000","1447813471104000","5465000026886400","14729684037868800","29321976905020800","43894642462070400","49742107587572400","42522503128625520","27023632735259160","12391849614176760","3878879027180430","742353680124630","65573803186921"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","37","361","4361","62701","1044205","19748177","417787921","9770678101","250194150581","6959638411705","208919770666777","6729933476435261","231512615111396221"],"nonn":true,"definitions":[{"position":0,"source":"derived:mobius_x_over_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["1","1"]]},"terms":[{"coeff":{"coeffs":[["1","4","4","1"],["1","1","-1","-1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","3","3","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":181558,"position":0,"claim":"real_rooted","from_index":0,"region_key":"[-1, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]","region_exact":null,"edges":["e-36a97bf38573f7ca"]},{"id":181559,"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-36a97bf38573f7ca"]},{"id":181560,"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-36a97bf38573f7ca"]}],"region_exact":{},"incoming_edges":[{"id":"e-36a97bf38573f7ca","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-381741d6edcf9397","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":"fTransform","lean_declarations":["RealRooted.splits_fPolynomial"],"certificate_hash":"58a8b2648f358d1717f7c1dbfc32626111b7478ede5845ecb78fccbfeb0ffdd1","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-011d67b73ad91983","role":"sequence","name":"basis_map_A008277 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-260cf7e5837f4e61","kind":"candidate_equal","node_a_id":"n-381741d6edcf9397","node_b_id":"n-9773dd1e68071da9","fingerprint":null,"oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"reason":"no all-index equality certificate was found","rows_compared":15},"other_id":"n-9773dd1e68071da9","other":{"id":"n-9773dd1e68071da9","name":"reversal of Triangle read by rows, T(n, k) = Sum_{i=0..n} L'(n, n-i) * binomial(i, k), for k = 0..n-1.","first_index":0,"fingerprint":"f6-01d09cfb9e6b0947","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","3"],["6","18","13"]],"card_rows":[["1"],["2","3"],["6","18","13"],["24","108","156","73"]],"row_sums_preview":["1","5","37","361","4361","62701","1044205","19748177"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d55979c30b97975a","steps":["x \u21a6 x+1","reverse"],"edges":["e-be016f00370af8fd","e-faba2e9611a1fb68"],"text":"A089231 \u2192 x \u21a6 x+1 \u2192 reverse"},"headline":"A089231 \u2192 x \u21a6 x+1 \u2192 reverse","chain":null}},{"id":"l-271def8a92d685a2","kind":"fingerprint_related","node_a_id":"n-eb51f16fedb3c293","node_b_id":"n-381741d6edcf9397","fingerprint":"f6-01d09cfb9e6b0947","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-01d09cfb9e6b0947"},"other_id":"n-eb51f16fedb3c293","other":{"id":"n-eb51f16fedb3c293","name":"Triangle read by rows, T(n, k) = Sum_{i=0..n} L'(n, n-i) * binomial(i, k), for k = 0..n-1.","first_index":0,"fingerprint":"f6-01d09cfb9e6b0947","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","2"],["13","18","6"]],"card_rows":[["1"],["3","2"],["13","18","6"],["73","156","108","24"]],"row_sums_preview":["1","5","37","361","4361","62701","1044205","19748177"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":["A059374"],"derivation":{"root":"n-d55979c30b97975a","steps":["x \u21a6 x+1"],"edges":["e-be016f00370af8fd"],"text":"A089231 \u2192 x \u21a6 x+1"},"headline":"A059374: Triangle read by rows, T(n, k) = Sum_{i=0..n} L'(n, n-i) * binomial(i, k), for k = 0..n-1.","chain":"A089231 \u2192 x \u21a6 x+1"}},{"id":"l-4b2bdd310188d53d","kind":"fingerprint_related","node_a_id":"n-e7c705bd49501967","node_b_id":"n-381741d6edcf9397","fingerprint":"f6-01d09cfb9e6b0947","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-01d09cfb9e6b0947"},"other_id":"n-e7c705bd49501967","other":{"id":"n-e7c705bd49501967","name":"mobius_x_over_x_plus_one of Triangle read by rows: the unsigned Lah numbers T(n, k) = binomial(n-1, k-1)*n!/k! if n > 0 and k > 0, T(n, 0) = 0^n and otherwise 0, for n >= 0 and 0 <= k <= n.","first_index":1,"fingerprint":"f6-01d09cfb9e6b0947","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["0","1"],["0","2","3"]],"card_rows":[["1"],["0","1"],["0","2","3"],["0","6","18","13"]],"row_sums_preview":["1","1","5","37","361","4361","62701","1044205"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9e4edf560ed848c1","steps":["x \u21a6 x/(x+1)"],"edges":["e-aea8708c7b8a1d37"],"text":"A271703 \u2192 x \u21a6 x/(x+1)"},"headline":"A271703 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-578ff2fc2d4616c6","kind":"fingerprint_related","node_a_id":"n-381741d6edcf9397","node_b_id":"n-d4794c9fc15957a0","fingerprint":"f6-01d09cfb9e6b0947","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-01d09cfb9e6b0947"},"other_id":"n-d4794c9fc15957a0","other":{"id":"n-d4794c9fc15957a0","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) = binomial(n,k)*(n-1)!/(k-1)!.","first_index":1,"fingerprint":"f6-01d09cfb9e6b0947","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","3"],["6","18","13"]],"card_rows":[["1"],["2","3"],["6","18","13"],["24","108","156","73"]],"row_sums_preview":["1","5","37","361","4361","62701","1044205","19748177"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4a0fd05ab0621175","steps":["x \u21a6 x/(x+1)"],"edges":["e-19b644289d9635b7"],"text":"A105278 \u2192 x \u21a6 x/(x+1)"},"headline":"A105278 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-664499252084a68a","kind":"fingerprint_related","node_a_id":"n-381741d6edcf9397","node_b_id":"n-471b3d369e427d51","fingerprint":"f6-01d09cfb9e6b0947","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-01d09cfb9e6b0947"},"other_id":"n-471b3d369e427d51","other":{"id":"n-471b3d369e427d51","name":"derivative of Triangle read by rows, T(n, k) = Sum_{i=0..n} L'(n, n-i) * binomial(i, k), for k = 0..n-1.","first_index":0,"fingerprint":"f6-01d09cfb9e6b0947","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["2"],["18","12"]],"card_rows":[["2"],["18","12"],["156","216","72"],["1460","3120","2160","480"]],"row_sums_preview":["0","2","30","444","7220","130830","2633442","58475480"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d55979c30b97975a","steps":["x \u21a6 x+1","D"],"edges":["e-be016f00370af8fd","e-1cec68d084c763f6"],"text":"A089231 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A089231 \u2192 x \u21a6 x+1 \u2192 D","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,        3;\n  2:       6,       18,        13;\n  3:      24,      108,       156,        73;\n  4:     120,      720,      1560,      1460,        501;\n  5:     720,     5400,     15600,     21900,      15030,       4051;\n  6:    5040,    45360,    163800,    306600,     315630,     170142,      37633;\n  7:   40320,   423360,   1834560,   4292400,    5891760,    4763976,    2107448,     394353;\n  8:  362880,  4354560,  22014720,  61810560,  106051680,  114335424,   75868128,   28393416,   4596553;\n  9: 3628800, 48988800, 283046400, 927158400, 1908930240, 2572547040, 2276043840, 1277703720, 413689770, 58941091;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (n+1+(n+4)*x-(n-4)*x^2-(n-1)*x^3)*P(n-1,x) + (x+3*x^2+3*x^3+x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((n + 1 + (n + 4)*x - (n - 4)*x^2 - (n - 1)*x^3)*P[n - 1, x] + (x + 3*x^2 + 3*x^3 + x^4)*D[P[n - 1, x], x])/(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 + 1 + (n + 4)*x - (n - 4)*x^2 - (n - 1)*x^3)*v[i - 1] + (x + 3*x^2 + 3*x^3 + x^4)*deriv(v[i - 1], x))/(1 + x)); 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 A008277","x \u21a6 x/(x+1)"],"edges":["e-31e8b29857f291e5","e-36a97bf38573f7ca"],"text":"A130534 \u2192 basis A008277 \u2192 x \u21a6 x/(x+1)"},"headline":"A130534 \u2192 basis A008277 \u2192 x \u21a6 x/(x+1)","chain":null}
