{"id":"n-264767197e69db61","revision":1,"first_index":1,"fingerprint":"f6-9c5a3526c6b32cef","name":"one_plus_derivative of Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["3","2"],["6","13","4"],["10","41","40","8"],["15","96","173","108","16"],["21","190","523","594","272","32"],["28","337","1286","2201","1816","656","64"],["36","553","2758","6465","7968","5152","1536","128"],["45","856","5362","16248","27209","26120","13872","3520","256"],["55","1266","9678","36468","78371","101650","79792","35936","7936","512"],["66","1805","16476","75078","199294","330005","348540","231216","90368","17664","1024"],["78","2497","26752","144342","460482","937305","1258920","1120072","643072","221952","38912","2048"],["91","3368","41767","262460","985721","2400520","3948965","4454548","3421280","1730816","534784","84992","4096"],["105","4446","63089","455598","1982123","5658770","11099355","15291234","14854240","10031808","4535040","1268224","184320","8192"],["120","5761","92638","760383","3782636","12460175","28558830","46785905","55380080","47211360","28442240","11619584","2967552","397312","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","23","99","408","1632","6388","24596","93488","351664","1311536","4856432","17873408","65436544","238480960"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0","-6","-8","4"],["1","6","12","-4"],["-2","0","-4","0"],["1","0","0","0"]]},"terms":[{"coeff":{"coeffs":[["0","-2","-12","12","-8"],["-1","0","20","-12","0"],["0","6","-4","0","0"],["1","0","0","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-4","-4","-8","-8"],["0","-2","-2","-8","-8","0"],["0","2","2","0","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":164187,"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-a682124e3a957353"]},{"id":164188,"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-a682124e3a957353"]},{"id":164189,"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-a682124e3a957353"]}],"region_exact":{},"incoming_edges":[{"id":"e-a682124e3a957353","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-264767197e69db61","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":"5919f55ee9b264c3aac359e98062a453f4754a2c186b292404addc6feb1f4a39","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-2821c21537bbbcdd","role":"sequence","name":"Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= 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-69fc20ed1b9155ec","kind":"oeis_match","node_a_id":"n-264767197e69db61","node_b_id":null,"fingerprint":null,"oeis_id":"A181331","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A181331","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A181331","relation":"match_row_sums"}],"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:  1;\n   2:  3,    2;\n   3:  6,   13,    4;\n   4: 10,   41,   40,     8;\n   5: 15,   96,  173,   108,    16;\n   6: 21,  190,  523,   594,   272,     32;\n   7: 28,  337, 1286,  2201,  1816,    656,    64;\n   8: 36,  553, 2758,  6465,  7968,   5152,  1536,   128;\n   9: 45,  856, 5362, 16248, 27209,  26120, 13872,  3520,  256;\n  10: 55, 1266, 9678, 36468, 78371, 101650, 79792, 35936, 7936, 512;\n","recurrence_sentence":"The row polynomials satisfy (n^3-2*n^2+n+(6*n-6)*x-(4*n^2-12*n+8)*x^2-(4*n-4)*x^3)*P(n,x) = (n^3-n+(6*n^2-2)*x-(4*n^2-20*n+12)*x^2-(12*n-12)*x^3-8*x^4)*P(n-1,x) + ((2*n^2-2*n)*x+(2*n^2-2*n-4)*x^2-(8*n+4)*x^3-(8*n+8)*x^4-8*x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((n^3 - n + (6*n^2 - 2)*x - (4*n^2 - 20*n + 12)*x^2 - (12*n - 12)*x^3 - 8*x^4)*P[n - 1, x] + ((2*n^2 - 2*n)*x + (2*n^2 - 2*n - 4)*x^2 - (8*n + 4)*x^3 - (8*n + 8)*x^4 - 8*x^5)*D[P[n - 1, x], x])/(n^3 - 2*n^2 + n + (6*n - 6)*x - (4*n^2 - 12*n + 8)*x^2 - (4*n - 4)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((n^3 - n + (6*n^2 - 2)*x - (4*n^2 - 20*n + 12)*x^2 - (12*n - 12)*x^3 - 8*x^4)*v[i - 1] + ((2*n^2 - 2*n)*x + (2*n^2 - 2*n - 4)*x^2 - (8*n + 4)*x^3 - (8*n + 8)*x^4 - 8*x^5)*deriv(v[i - 1], x))/(n^3 - 2*n^2 + n + (6*n - 6)*x - (4*n^2 - 12*n + 8)*x^2 - (4*n - 4)*x^3)); 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-1bf99ad40f056d33","steps":["basis A056242","D+1"],"edges":["e-3cb5f3d64c4bb317","e-a682124e3a957353"],"text":"basis-map coefficients of A026729 \u2192 basis A056242 \u2192 D+1"},"headline":"basis-map coefficients of A026729 \u2192 basis A056242 \u2192 D+1","chain":null}
