{"id":"n-4e47c45c71c4060b","revision":0,"first_index":0,"fingerprint":"f6-269631c8dfab89fe","name":"derivative of Interpolating the factorial and the powers of 2. Triangle read by rows, T(n, k) for 0 <= 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":[[],["2"],["8","8"],["36","72","24"],["192","576","384","64"],["1200","4800","4800","1600","160"],["8640","43200","57600","28800","5760","384"],["70560","423360","705600","470400","141120","18816","896"],["645120","4515840","9031680","7526400","3010560","602112","57344","2048"],["6531840","52254720","121927680","121927680","60963840","16257024","2322432","165888","4608"],["72576000","653184000","1741824000","2032128000","1219276800","406425600","77414400","8294400","460800","10240"],["878169600","8781696000","26345088000","35126784000","24588748800","9835499520","2341785600","334540800","27878400","1239040","22528"],["11496038400","126456422400","421521408000","632282112000","505825689600","236051988480","67443425280","12043468800","1338163200","89210880","3244032","49152"],["161902540800","1942830489600","7123711795200","11872852992000","10685567692800","5698969436160","1899656478720","407069245440","56537395200","5025546240","274120704","8306688","106496"],["2440992153600","31732897996800","126931591987200","232707918643200","232707918643200","139624751185920","53190381404160","13297595351040","2216265891840","246251765760","17909219328","814055424","20873216","229376"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","2","16","132","1216","12560","144384","1830752","25391104","382355712","6211594240","108261452288","2014551220224","39854406146048","835115309400064"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["2"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0"],["0","2"],["1","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0"],["0","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":182832,"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-4ff7b2e3f96d0d7e"]},{"id":182833,"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-4ff7b2e3f96d0d7e"]},{"id":182834,"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-4ff7b2e3f96d0d7e"]}],"region_exact":{},"incoming_edges":[{"id":"e-4ff7b2e3f96d0d7e","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-4e47c45c71c4060b","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":"4f1982415840e8f5770ece4002778d62a757e7bee62344eb4891b3c44dd46c80","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-8b66c9f49e6d2f0a","role":"sequence","name":"Interpolating the factorial and the powers of 2. Triangle read by rows, T(n, k) for 0 <= k <= n.","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-17bcbd5aa18d9331","kind":"fingerprint_related","node_a_id":"n-cbd7c2e66114e926","node_b_id":"n-4e47c45c71c4060b","fingerprint":"f6-269631c8dfab89fe","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-269631c8dfab89fe"},"other_id":"n-cbd7c2e66114e926","other":{"id":"n-cbd7c2e66114e926","name":"Triangle read by rows: T(n,k) = 2^k * (n!/k!)*binomial(n-1,k-1).","first_index":0,"fingerprint":"f6-269631c8dfab89fe","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["2"],["4","4"],["12","24","8"]],"card_rows":[["2"],["4","4"],["12","24","8"],["48","144","96","16"]],"row_sums_preview":["2","8","44","304","2512","24064","261536","3173888"],"regions":["(-inf, 0]","(-inf, 0)"],"region_exact":{},"oeis_ids":["A086915"],"derivation":{"root":"n-cbd7c2e66114e926","steps":[],"edges":[],"text":"A086915"},"headline":"A086915: Triangle read by rows: T(n,k) = 2^k * (n!/k!)*binomial(n-1,k-1).","chain":null}},{"id":"l-80e2f1a1003fea63","kind":"fingerprint_related","node_a_id":"n-4e47c45c71c4060b","node_b_id":"n-c4d9ef50e4651641","fingerprint":"f6-269631c8dfab89fe","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-269631c8dfab89fe"},"other_id":"n-c4d9ef50e4651641","other":{"id":"n-c4d9ef50e4651641","name":"reversal of Triangle read by rows: T(n,k) = 2^k * (n!/k!)*binomial(n-1,k-1).","first_index":0,"fingerprint":"f6-269631c8dfab89fe","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["2"],["4","4"],["8","24","12"]],"card_rows":[["2"],["4","4"],["8","24","12"],["16","96","144","48"]],"row_sums_preview":["2","8","44","304","2512","24064","261536","3173888"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cbd7c2e66114e926","steps":["reverse"],"edges":["e-adb9a693c050b6f8"],"text":"A086915 \u2192 reverse"},"headline":"A086915 \u2192 reverse","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:       0;\n  1:       2;\n  2:       8,        8;\n  3:      36,       72,        24;\n  4:     192,      576,       384,        64;\n  5:    1200,     4800,      4800,      1600,      160;\n  6:    8640,    43200,     57600,     28800,     5760,      384;\n  7:   70560,   423360,    705600,    470400,   141120,    18816,     896;\n  8:  645120,  4515840,   9031680,   7526400,  3010560,   602112,   57344,   2048;\n  9: 6531840, 52254720, 121927680, 121927680, 60963840, 16257024, 2322432, 165888, 4608;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (n^2+2*n*x)*P(n-1,x) + n*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((n^2 + 2*n*x)*P[n - 1, x] + n*x*D[P[n - 1, x], x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((n^2 + 2*n*x)*v[i - 1] + n*x*deriv(v[i - 1], x))/(n - 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-8b66c9f49e6d2f0a","steps":["D"],"edges":["e-4ff7b2e3f96d0d7e"],"text":"A331333 \u2192 D"},"headline":"A331333 \u2192 D","chain":null}
