{"id":"n-c0db4c865e1a9a1a","revision":0,"first_index":1,"fingerprint":"f6-ced16c5d17801fb2","name":"derivative of Triangle T = A007318*A271703; T(n,m)= Sum_{i=0..n} L'(n,i)*binomial(i,m), m=0..n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],["1"],["4","2"],["21","18","3"],["136","156","48","4"],["1045","1460","630","100","5"],["9276","15030","8160","1860","180","6"],["93289","170142","109725","32060","4515","294","7"],["1047376","2107448","1558368","549080","100240","9576","448","8"],["12975561","28393416","23508828","9616824","2137590","266616","18396","648","9"],["175721140","413689770","377055360","174531000","45417960","6941340","628320","32760","900","10"],["2581284541","6483520010","6422902695","3302377320","979557810","176274252","19637310","1347720","54945","1210","11"],["40864292184","108777654612","115975952400","65320138620","21661825680","4466229768","588196224","49836600","2680920","87780","1584","12"],["693347907421","1945345308348","2214742136178","1351724262460","493894184355","114381733752","17323382076","1740240216","115836435","5013580","134706","2028","13"],["12548540320876","36941705020766","44623807064928","29259602976604","11645553499580","2984247988722","509418061152","59010127176","4667130468","250380130","8904896","199836","2548","14"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","1","6","42","344","3240","34512","410032","5372544","76917888","1194018560","19966957824","357706896384","6832620141568","138576811677696"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["1","-1"],["-2","1"],["1","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1","-1"],["1","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":181561,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-71ee257a063243ca"]},{"id":181562,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-71ee257a063243ca"]},{"id":181563,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-71ee257a063243ca"]}],"region_exact":{},"incoming_edges":[{"id":"e-71ee257a063243ca","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-c0db4c865e1a9a1a","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"7202414d59eba79761eec24bc77ba50243cc1cdd5aab343e8f8d24162e8a1bf7","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-01897e65d9bb3867","role":"sequence","name":"Triangle T = A007318*A271703; T(n,m)= Sum_{i=0..n} L'(n,i)*binomial(i,m), m=0..n.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-63d1fbf263ccd383","kind":"fingerprint_related","node_a_id":"n-148a4e95b74cc241","node_b_id":"n-c0db4c865e1a9a1a","fingerprint":"f6-ced16c5d17801fb2","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-ced16c5d17801fb2"},"other_id":"n-148a4e95b74cc241","other":{"id":"n-148a4e95b74cc241","name":"mobius_x_over_x_plus_one of T(n,k) is the number of partial bijections of height k (height(alpha) = |Im(alpha)|) of an n-element set.","first_index":0,"fingerprint":"f6-ced16c5d17801fb2","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["1","6","7"]],"card_rows":[["1"],["1","2"],["1","6","7"],["1","12","39","34"]],"row_sums_preview":["1","3","14","86","648","5752","58576","671568"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-47fe395e9f7d5f86","steps":["x \u21a6 x/(x+1)"],"edges":["e-6a2c9a503d8f313d"],"text":"A144084 \u2192 x \u21a6 x/(x+1)"},"headline":"A144084 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-7bbeaeccdab2f637","kind":"fingerprint_related","node_a_id":"n-c0db4c865e1a9a1a","node_b_id":"n-d04cf1089b783b9c","fingerprint":"f6-ced16c5d17801fb2","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-ced16c5d17801fb2"},"other_id":"n-d04cf1089b783b9c","other":{"id":"n-d04cf1089b783b9c","name":"one_plus_derivative of Triangle T = A007318*A271703; T(n,m)= Sum_{i=0..n} L'(n,i)*binomial(i,m), m=0..n.","first_index":1,"fingerprint":"f6-ced16c5d17801fb2","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["7","6","1"]],"card_rows":[["1"],["2","1"],["7","6","1"],["34","39","12","1"]],"row_sums_preview":["1","3","14","86","648","5752","58576","671568"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-01897e65d9bb3867","steps":["D+1"],"edges":["e-271e415dbcb3a2ae"],"text":"A059110 \u2192 D+1"},"headline":"A059110 \u2192 D+1","chain":null}},{"id":"l-ae4aff27e2b03546","kind":"fingerprint_related","node_a_id":"n-b9a589cc34ffc178","node_b_id":"n-c0db4c865e1a9a1a","fingerprint":"f6-ced16c5d17801fb2","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-ced16c5d17801fb2"},"other_id":"n-b9a589cc34ffc178","other":{"id":"n-b9a589cc34ffc178","name":"shift_x_plus_one of basis_map_A008277 of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.","first_index":0,"fingerprint":"f6-ced16c5d17801fb2","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["7","6","1"]],"card_rows":[["1"],["2","1"],["7","6","1"],["34","39","12","1"]],"row_sums_preview":["1","3","14","86","648","5752","58576","671568"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b9fbfb57d9269e2f","steps":["basis A008277","x \u21a6 x+1"],"edges":["e-07cfebc08bea7e3c","e-8357b32e6e3ab54f"],"text":"A132393 \u2192 basis A008277 \u2192 x \u21a6 x+1"},"headline":"A132393 \u2192 basis A008277 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-cace9596e54b8c5c","kind":"fingerprint_related","node_a_id":"n-c0db4c865e1a9a1a","node_b_id":"n-184f8d63c8db0227","fingerprint":"f6-ced16c5d17801fb2","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-ced16c5d17801fb2"},"other_id":"n-184f8d63c8db0227","other":{"id":"n-184f8d63c8db0227","name":"shift_x_plus_one of Triangle of coefficients of n!*(1 - x)^n*L_n(x/(1 - x)), where L_n(x) is the Laguerre polynomial.","first_index":1,"fingerprint":"f6-ced16c5d17801fb2","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":0,"preview_rows":[["1"],["-1","-2"],["1","6","7"]],"card_rows":[["1"],["-1","-2"],["1","6","7"],["-1","-12","-39","-34"]],"row_sums_preview":["1","-3","14","-86","648","-5752","58576","-671568"],"regions":["(-1, 0)"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-db839974123087ee","steps":["x \u21a6 x+1"],"edges":["e-f386eb84f6c9d238"],"text":"A123202 \u2192 x \u21a6 x+1"},"headline":"A123202 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-d1ebfd55369431d7","kind":"fingerprint_related","node_a_id":"n-c0db4c865e1a9a1a","node_b_id":"n-18383b729ceef751","fingerprint":"f6-ced16c5d17801fb2","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-ced16c5d17801fb2"},"other_id":"n-18383b729ceef751","other":{"id":"n-18383b729ceef751","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n, k) = 2^n * [x^k] hypergeom([-n, -n, 1], [1], x).","first_index":1,"fingerprint":"f6-ced16c5d17801fb2","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["4","24","28"],["8","96","312","272"],["16","320","2016","4672","3344"]],"card_rows":[["4","24","28"],["8","96","312","272"],["16","320","2016","4672","3344"],["32","960","9920","43520","80160","49472"]],"row_sums_preview":["56","688","10368","184064","3748864","85960704","2187886592","61133750272"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2ee8dc15b8ca5b2d","steps":["x \u21a6 x/(x+1)"],"edges":["e-c6b10d6190de873e"],"text":"A395972 \u2192 x \u21a6 x/(x+1)"},"headline":"A395972 \u2192 x \u21a6 x/(x+1)","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:        0;\n   2:        1;\n   3:        4,        2;\n   4:       21,       18,        3;\n   5:      136,      156,       48,       4;\n   6:     1045,     1460,      630,     100,       5;\n   7:     9276,    15030,     8160,    1860,     180,      6;\n   8:    93289,   170142,   109725,   32060,    4515,    294,     7;\n   9:  1047376,  2107448,  1558368,  549080,  100240,   9576,   448,   8;\n  10: 12975561, 28393416, 23508828, 9616824, 2137590, 266616, 18396, 648, 9;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = (n^2-2*n+1+(n-1)*x)*P(n-1,x) + (n-1+(n-1)*x)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((n^2 - 2*n + 1 + (n - 1)*x)*P[n - 1, x] + (n - 1 + (n - 1)*x)*D[P[n - 1, x], x])/(n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((n^2 - 2*n + 1 + (n - 1)*x)*v[i - 1] + (n - 1 + (n - 1)*x)*deriv(v[i - 1], x))/(n - 2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-01897e65d9bb3867","steps":["D"],"edges":["e-71ee257a063243ca"],"text":"A059110 \u2192 D"},"headline":"A059110 \u2192 D","chain":null}
